It has been generally accepted for about 2000 years that Acts was written before 70 a.d. There is no mention of the destruction of the temple, nor the death of Paul or any of the Apostles except Jacob, the son of Zebedee (Acts 12:2). I find it interesting that Jacob's murder is included, yet the author fails to mention the events of Peter, Jacob the just, or Paul's death. In fact, the author doesn't even indicate any of these people are dead. At the end, all that's said is:
"And Paul dwelt two whole years in his own hired house, and received all that came in unto him,
Preaching the kingdom of God, and teaching those things which concern the Lord Jesus Christ, with all confidence, no man forbidding him." Acts 28:30-31
This doesn't tell us anything. Was Paul killed two years later? Did somebody begin forbidding Paul to preach his gospel after two years? The churches of Asia perhaps (2 Timothy 1:15)?
Also significant is the authors use of "we" several times throughout Acts. If this were any other book, there would be no question as to when it was written. That is my own opinion of course, but here is a link to a lengthy excerpt of John A.T. Robinson's book Redating the New Testament for anyone interested in learning more about the validity of Acts and the other NT writings. Also please see What are the arguments that Acts was written prior to 70 AD? and Which NT books were written after the destruction of the temple? for additional information.
This assumption that the writer of Acts (from this time on I'll refer to this person as Luke) was mistaken seems to be taken exclusively on the idea that Paul was not lying in Galatians. This is understandable, since Paul says:
"Now the things which I write unto you, behold, before God, I lie not." Galatians 1:20
So if one wishes to take Paul's word as infallible, then I guess you have your proof right there. However, I find it highly unlikely that Luke was either mistaken or "exercising literary license".
It seems much more plausible that Luke was quite aware that he was reporting conflicting events. Based on the amount of detail concerning Paul's experience outside of Damascus and the events that followed, Luke writes chapter 9 as though it's what was directly told to him by Paul. I'll quote it so we can compare:
"And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,
And desired of him letters to Damascus to the synagogues, that if he found any of this way, whether they were men or women, he might bring them bound unto Jerusalem.
And as he journeyed, he came near Damascus: and suddenly there shined round about him a light from heaven:
And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me?
And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest: it is hard for thee to kick against the pricks.
[And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him,] (Most translations omit this) Arise, and go into the city, and it shall be told thee what thou must do.
And the men which journeyed with him stood speechless, hearing a voice (φωνῆς: sound, voice) , but seeing no man.
And Saul arose from the earth; and when his eyes were opened, he saw no man: but they led him by the hand, and brought him into Damascus.
And he was three days without sight, and neither did eat nor drink.
And there was a certain disciple at Damascus, named Ananias; and to him said the Lord in a vision, Ananias. And he said, Behold, I am here, Lord.
And the Lord said unto him, Arise, and go into the street which is called Straight, and inquire in the house of Judas for one called Saul, of Tarsus: for, behold, he prayeth,
And hath seen in a vision a man named Ananias coming in, and putting his hand on him, that he might receive his sight.
Then Ananias answered, Lord, I have heard by many of this man, how much evil he hath done to thy saints at Jerusalem:
And here he hath authority from the chief priests to bind all that call on thy name.
But the Lord said unto him, Go thy way: for he is a chosen vessel unto me, to bear my name before the Gentiles, and kings, and the children of Israel:
For I will shew him how great things he must suffer for my name's sake.
And Ananias went his way, and entered into the house; and putting his hands on him said, Brother Saul, the Lord, even Jesus, that appeared unto thee in the way as thou camest, hath sent me, that thou mightest receive thy sight, and be filled with the Holy Ghost.
And immediately there fell from his eyes as it had been scales: and he received sight forthwith, and arose, and was baptized.
And when he had received meat, he was strengthened. Then was Saul certain days with the disciples which were at Damascus." Acts 9:1-19
So in this account, Paul saw "a light" and heard a voice. The men with Paul did not witness anything, but they heard a voice/sound. He is then taken to Ananias, who had a vision of 'the Lord' telling him a few words. Ananias puts his hands on Paul, says a few words, and immediately scales fall from his eyes.
When we get to Acts 21, we see Paul was warned beforehand not to go to Jerusalem. He honorably says he is ready to die for the Lord:
"And when he was come unto us, he took Paul's girdle, and bound his own hands and feet, and said, Thus saith the Holy Ghost, So shall the Jews at Jerusalem bind the man that owneth this girdle, and shall deliver him into the hands of the Gentiles.
And when we heard these things, both we, and they of that place, besought him not to go up to Jerusalem.
Then Paul answered, What mean ye to weep and to break mine heart? for I am ready not to be bound only, but also to die at Jerusalem for the name of the Lord Jesus." Acts 21:11-13
However, Paul was obviously not aware of what would happen next. The Jews were very angry at Paul. There was a man, named Trophimus (an Ephesian) that was seen with Paul earlier that day. Trophimus was an uncircumcised gentile, and he had gone into the temple- which was forbidden:
"But let none come into the house of YHVH, save the priests, and they that minister of the Levites; they shall go in, for they are holy: but all the people shall keep the watch of YHVH." 2 Chronicles 23:6
"Thus saith the Lord YHVH; No stranger, uncircumcised in heart, nor uncircumcised in flesh, shall enter into my sanctuary, of any stranger that is among the children of Israel." Ezekiel 44:9
So the Jews were pretty freaked out. They began accusing Paul of teaching all men everywhere against the Jewish people, the law, and the temple:
"And when the seven days were almost ended, the Jews which were of Asia, when they saw him in the temple, stirred up all the people, and laid hands on him,
Crying out, Men of Israel, help: This is the man, that teacheth all men every where against the people, and the law, and this place: and further brought Greeks also into the temple, and hath polluted this holy place.
(For they had seen before with him in the city Trophimus an Ephesian, whom they supposed that Paul had brought into the temple.)
And all the city was moved, and the people ran together: and they took Paul, and drew him out of the temple: and forthwith the doors were shut.
And as they went about to kill him, tidings came unto the chief captain of the band, that all Jerusalem was in an uproar." Acts 21:27-31
The Jews were about to kill Paul until the chief captain was told that all of Jerusalem was angry. He couldn't figure out what Paul had done because so many people were shouting at once, so he led Paul away. Paul then asks if he may present his case to the Jews. This is what he says:
And it came to pass, that, as I made my journey, and was come nigh unto Damascus about noon, suddenly there shone from heaven a great light round about me.
And I fell unto the ground, and heard a voice saying unto me, Saul, Saul, why persecutest thou me?
And I answered, Who art thou, Lord? And he said unto me, I am Jesus of Nazareth, whom thou persecutest.
And they that were with me saw indeed the light, and were afraid; but they heard not the voice of him that spake to me.
And I said, What shall I do, Lord? And the Lord said unto me, Arise, and go into Damascus; and there it shall be told thee of all things which are appointed for thee to do.
And when I could not see for the glory of that light, being led by the hand of them that were with me, I came into Damascus.
And one Ananias, a devout man according to the law, having a good report of all the Jews which dwelt there,
Came unto me, and stood, and said unto me, Brother Saul, receive thy sight. And the same hour I looked up upon him.
And he said, The God of our fathers hath chosen thee, that thou shouldest know his will, and see that Just One, and shouldest hear the voice of his mouth.
For thou shalt be his witness unto all men of what thou hast seen and heard.
And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord." Acts 22:6-16
In this account, Paul again sees a light. The men that are with him suddenly become "witnesses" to the light, but they don't hear any sound. Paul is taken to Ananius, who doesn't touch Paul, but simply tells him to receive his sight. Rather than immediately, it's now "within the hour" that Paul can see. Ananius then tells Paul a bunch of stuff that is not anything similar to what "the Lord" had said in Acts 9.
The Jews, upon hearing this, think like most people would when they hear a man say he saw a light and heard voices. They thought Paul was insane. So the chief captain took Paul aside, and they had decided to scourge him. This was going to be painful, so rather than suffer, Paul appeals to being a Roman citizen. The Romans decide not to harm him, and the next day he appeals to the Jews again. The high priest has his men smack Paul in the mouth, to which Paul replies:
"Then said Paul unto him, God shall smite thee, thou whited wall: for sittest thou to judge me after the law, and commandest me to be smitten contrary to the law?
And they that stood by said, Revilest thou God's high priest?
Then said Paul, I wist not, brethren, that he was the high priest: for it is written, Thou shalt not speak evil of the ruler of thy people." Acts 23:3-5
Then Paul realizes that some of the Jews are Sadducees and others are Pharisees. The Pharisees believe in a resurrection of the dead, so Paul (being a Pharisee) appeals to them. He then says something very interesting:
"But when Paul perceived that the one part were Sadducees, and the other Pharisees, he cried out in the council, Men and brethren, I am a Pharisee, the son of a Pharisee: of the hope and resurrection of the dead I am called in question." Acts 23:6
We just saw that the Jews were angry because they accused Paul of teaching against the law and bringing a Gentile into the temple. But Paul says they're angry because he was preaching the resurrection of the dead. Is this another mistake on Luke's part? I doubt it.
So some Jews make a vow to kill Paul, and when the chief captain hears about this, he sends Paul to the governor Felix. Felix keeps Paul bound for two years, until Festus enters the picture. The Jews tell Festus what they have against Paul, and Festus decides that Paul should be sent to Jerusalem to be tried by his own people. Remember when Paul said he was ready to die in Jerusalem for the Lord? Well, he must have had a change of heart:
"But Festus, willing to do the Jews a pleasure, answered Paul, and said, Wilt thou go up to Jerusalem, and there be judged of these things before me?
Then said Paul, I stand at Caesar's judgment seat, where I ought to be judged: to the Jews have I done no wrong, as thou very well knowest.
For if I be an offender, or have committed any thing worthy of death, I refuse not to die: but if there be none of these things whereof these accuse me, no man may deliver me unto them. I appeal unto Caesar. Acts 25:9-11
So next Paul is sent to king Agrippa, and this is when he tells his conversion story for a second time:
"Whereupon as I went to Damascus with authority and commission from the chief priests,
At midday, O king, I saw in the way a light from heaven, above the brightness of the sun, shining round about me and them which journeyed with me.
And when we were all fallen to the earth, I heard a voice speaking unto me, and saying in the Hebrew tongue, Saul, Saul, why persecutest thou me? it is hard for thee to kick against the pricks.
And I said, Who art thou, Lord? And he said, I am Jesus whom thou persecutest.
But rise, and stand upon thy feet: for I have appeared unto thee for this purpose, to make thee a minister and a witness both of these things which thou hast seen, and of those things in the which I will appear unto thee;
Delivering thee from the people, and from the Gentiles, unto whom now I send thee,
To open their eyes, and to turn them from darkness to light, and from the power of Satan unto God, that they may receive forgiveness of sins, and inheritance among them which are sanctified by faith that is in me." Acts 26:12
That is a much different story than any of the other two. This time, Paul doesn't even mention Ananius. Instead he says this Jesus spoke directly to him, telling him the entire mission he was to accomplish.
All three of these accounts are way too inconsistent to be a mistake or considered "literary licence". There's no doubt that Luke loved Paul, but he wasn't going to lie for him. Luke knew the truth, and according to Albert W. Pink, "Luke's Gospel is concerned with the Humanity of our Lord".
So then we get to Paul's own account of his conversion in Galatians 1. He starts the letter off with:
"Paul, an apostle, (not of men, neither by man, but by Jesus Christ, and God the Father, who raised him from the dead;)" Galatians 1:1
Paul says he was sent out (an apostle), but not by man: he claims to be an apostle of Yeshua himself. Rather than quote them, the following verses show Paul being "sent out" several times by men: Acts 9:30, 11:30, 14:14, 15:22...
Paul then tells the Galatians that they should not listen to any other gospel than the one he has already told them:
"I marvel that ye are so soon removed from him that called you into the grace of Christ unto another gospel.
Which is not another; but there be some that trouble you, and would pervert the gospel of Christ.
But though we, or an angel from heaven, preach any other gospel unto you than that which we have preached unto you, let him be accursed.
As we said before, so say I now again, If any man preach any other gospel unto you than that ye have received, let him be accursed." Galatians 1:6-9
Notice Paul says "we" as though to say "me or any of the 'other' Apostles". It seems the Galatians had been hearing a different gospel, probably from Peter and John since Paul sets out to discredit them in the next chapter. Then Paul says:
"But I certify you, brethren, that the gospel which was preached of me is not after man.
For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.
For ye have heard of my conversation in time past in the Jews' religion, how that beyond measure I persecuted the church of God, and wasted it:
And profited in the Jews' religion above many my equals in mine own nation, being more exceedingly zealous of the traditions of my fathers.
But when it pleased God, who separated me from my mother's womb, and called me by his grace,
To reveal his Son in me, that I might preach him among the heathen; immediately I conferred not with flesh and blood:
Neither went I up to Jerusalem to them which were apostles before me; but I went into Arabia, and returned again unto Damascus.
Then after three years I went up to Jerusalem to see Peter, and abode with him fifteen days.
But other of the apostles saw I none, save James the Lord's brother.
Now the things which I write unto you, behold, before God, I lie not." Galatians 1:11-20
Paul says that he did not learn his gospel from any man. Instead, it was by revelation of Jesus Christ, delivered to Paul in the secret chambers of his own mind. He then boasts about how he worked zealously to persecute the church because of his faithfulness to his prior religion, but God set him apart. "Immediately" he did not talk to any men about his revelation, nor the Apostles "before him" (again signifying that he is one of the twelve). Instead he went to Arabia for three years. Luke makes no mention of Arabia, and shows Paul moving from Damascus straight to Jerusalem. Paul ends his conversion account with "before God, I lie not".
Acts was written before 70 a.d. and before the death of Peter and Paul. The author presents us with three different accounts of Paul's conversion that are so different, it is highly unlikely that this was a simple mistake. Paul lied to the Galatians because his gospel was losing it's authority, so he tried to distinguish himself from the real Apostles by claiming to be a direct mediator between man and Jesus Christ. YHVH the only true God, father of Yeshua and all his brothers, has stated many times that He will send false prophets to test us. It sucks, because I've always loved Paul. However, YHVH is our only father, and our master Yeshua teaches us everything we need to know.
Per request for a demonstration that Paul would stoop this low, please see my answer to What role did James have in Paul's conflict with Peter?