Personally I don't believe Damascus to have been in Arabia. When Paul got saved, he was on his way to Damascus to persecute Christians. The Christians were scattered throughout Judaea and Samaria, but not outside of Israel:
Acts 8:1 (KJV)
… And at that time there was a great persecution against the church which was at Jerusalem; and they were all scattered abroad throughout the regions of Judaea and Samaria…
If Damascus wasn't part of Israel, Saul technically couldn't have done anything there, but Damascus belonged to Judah.
2 Kings 14:28 (KJV)
Now the rest of the acts of Jeroboam, and all that he did, and his might, how he warred, and how he recovered Damascus, and Hamath, which belonged to Judah, for Israel, are they not written in the book of the chronicles of the kings of Israel?
So, I don't believe that Damascus was in Arabia. Other than that, the Scriptures clearly state that Paul went to Arabia, then returned to Damascus, then left Damascus after three years.
Galatians 1:17-18 (KJV)
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.
So, one would have to ask "Where did Paul return to, Arabia or Damascus?" If it was Arabia, then there might be some scriptural proof of this. I cannot find any other place in scripture where Paul was in Arabia (other than by saying that Damascus is in Arabia, which is not scriptural proof). If it is in reference to Damascus, then one need not look further than Acts 9 to find a time that he was in Damascus:
Acts 9:8 (KJV)
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…
So, Jesus appeared to Saul while he was on his way to persecute Christians in Damascus, when He blinded Saul, and told him to go into Damascus where he would receive further instruction. The men then guided Saul into Damascus to the house of Judas (Acts 9:11). Ananias found him there and returned Saul's sight to him. Now it is not clear how long he was in Damascus before he left for Arabia, but he was there for some time:
Acts 9:19,22-23 (KJV)
19And when he had received meat, he was strengthened. Then was Saul certain days with the disciples which were at Damascus…
22 But Saul increased the more in strength, and confounded the Jews which dwelt at Damascus, proving that this is very Christ. 23And after that many days were fulfilled, the Jews took counsel to kill him…
He was there "certain days" and then "many days". I believe that these are two separate times. Three years could be described as many days (1,095 days).
Now:
Galatians 1:17-18 (KJV)
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.
Paul didn't go up to Jerusalem prior to the three-year period which is in question, but he eventually did:
Acts 9:26 (KJV)
And when Saul was come to Jerusalem, he assayed to join himself to the disciples: but they were all afraid of him, and believed not that he was a disciple.
So, the three-year period takes place between Acts 9:8 & Acts 9:26. I believe the timeline allows him to have left for Arabia after Acts 9:19.
Acts 9:19 (KJV)
And when he had received meat, he was strengthened. Then was Saul certain days with the disciples which were at Damascus.
He was with the disciples "certain days" at Damascus. It certainly doesn't sound like it was a long time as compared to "many days" four verses later (v. 23).
Therefore, he was saved in Acts 9:4-6, went to Damascus in Acts 9:8, had his sight returned and was baptized in Acts 9:18, he stayed with the disciples there for "certain days" in Acts 9:19, left for Arabia and shortly returned back to Damascus as stated in Ga 1:17, then immediately started preaching in Damascus in Acts 9:20, then after "many days" (read three years as stated in Galatians 1:18) the Jews decided to try to kill him in Acts 9:23, and then left for Jerusalem in Acts 9:26 as stated in Galatians 1:18.
It can be a little confusing because Acts 9:19 says he was with disciples in Damascus, but Galatians 1:17 says he didn't go to Jerusalem to those that were disciples before him. The issue isn't that he wasn't with any disciples that were before him at all, but that he wasn't in Jerusalem with any that were disciples before him.
Another timing possibility would be him leaving for Arabia while he was blind. It is necessary to examine a few verses together for this.
Acts 9:8-9 (KJV)
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.
Galatians 1:16-17 (KJV) 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.
If Galatians 1:16 follows the Lord appearing to him on the way to Damascus, when did "immediately I conferred not with flesh and blood" happen? It has been established that “returned again unto Damascus” means he had already been in Damascus (he had to have been there before to return back there). So he could have gone to Arabia immediately after the men led him into Damascus. It would have been, at maximum, three days, which could have been possible. While Acts 9:9 sounds like he was led into Damascus and stayed there for three days, Paul could have told them to lead him into Arabia, then return again unto Damascus. Remember, they were already on their way to Damascus, so it is only natural for the men to have led him the rest of the way there. Then once there, they could have left for somewhere else for a short period of time. This could be the case, however, in no way does this accommodate a three-year period in Arabia like many people believe. He was without food and sight for only three days, then he ate and was strengthened in Acts 9:19, and his sight was restored in Acts 9:18.
While it does work for him to have gone into Arabia for a very short time while he was blind, the men would have had to lead him into Damascus, then to Arabia, then back into Damascus. I don’t believe he would have done that, as he went into Damascus to be to told what to do as instructed by the Lord. He knew how to follow orders, and would have waited until he knew what he was supposed to do.
In Galatians 1:16 Paul stated that he didn’t confer with “flesh and blood”, simply stated, he didn’t talk to anybody about what to do, he just did what he was told to do. He would have stayed in Damascus till he was told what to do like the Lord said in Acts 9:6 - “… Arise, and go into the city, and it shall be told thee what thou must do”. Having said that, there is also the possibility that upon entering into Damascus someone could have told them to go into Arabia for a few days, and the bible just doesn’t mention it. If that is the case, then he would have done what he was told by the Lord, and didn’t confer with “flesh and blood” about it, as he would have done it without getting further advice nor talking about it. Ultimately having Ananias find him at the house of Judas:
Acts 9:11 (KJV)
And the Lord said unto him, Arise, and go into the street which is called Straight, and enquire in the house of Judas for one called Saul, of Tarsus: for, behold, he prayeth.
On one last note, you see further proof that Damascus is not in Arabia when you look at the progression once again with Galatians 1:16-17 in mind. Paul was blinded by the Lord, then the men led him into Damascus (Acts 9:8 says that he ...arose from the earth...and brought him into Damascus). They didn't wait around, or take him on a sight seeing tour, they immediately brought him into Damascus, (now this is where we come to Galatians 1:16-17) where he did not seek council on what he should do, but waited till instruction would be given to him, then he had to leave Damascus to go to Arabia. Had Damascus been in Arabia, then Paul would not have needed to say that went into Arabia, because he was next to Damascus, and would have already been in Arabia. Acts 9:3 says that they "...came near Damascus...", so we can confidently say that they were not in Arabia, nor close to it, as they were close to Damascus.
Whichever timeline is correct, a couple of things I believe are certain, which are that Paul never spent three years in Arabia, and that Damascus was in Israel. Pray about it, study it, and see what the Lord gives you.