I don't think so. The passage has been viewed in at least two ways. In both views the purpose of the words of Jesus were still to answer a trick question by the pharisees.
Matthew 22:15-22New International Version (NIV)
Paying the Imperial Tax to Caesar
15 Then the Pharisees went out and laid plans to trap him in his words. 16 They sent their disciples to him along with the Herodians. “Teacher,” they said, “we know that you are a man of integrity and that you teach the way of God in accordance with the truth. You aren’t swayed by others, because you pay no attention to who they are. 17 Tell us then, what is your opinion? Is it right to pay the imperial tax to Caesar or not?”
18 But Jesus, knowing their evil intent, said, “You hypocrites, why are you trying to trap me? 19 Show me the coin used for paying the tax.” They brought him a denarius, 20 and he asked them, “Whose image is this? And whose inscription?”
21 “Caesar’s,” they replied.
Then he said to them, “So give back to Caesar what is Caesar’s, and to God what is God’s.”
22 When they heard this, they were amazed. So they left him and went away.
This affirmed the governments role to collect taxes but still appeased the people who would have been upset by it in pointing to their true responsibility.
Some place that responsibility as the tithes and offerings should also still be given to God. The other view which happens to be mine is that Jesus is saying this coin bears the image of Caesar so he rules it's usage. The one that bears the image of God (mankind) should be given to God. So pay your taxes by giving the things produced by man to man's government. Give what is produced by God (yourself) to God's government, the kingdom of God.
In relating to "separation of church and state" I think the origin of the phrase should be looked at.
"The First Amendment has erected a wall of separation between church and state, but that wall is a one directional wall; it keeps the government from running the church, but it makes sure that Christian principles will always stay in government."
--Thomas Jefferson, President of the United States January 1, 1802 in an address to the Danbury Baptists.
The intention is clear that government should not be involved in matters of church, but that church should be involved in matters of government. This has of course been re-interpreted several times and in several ways but it's divergent to get into those points. The passage brought up is not a statement regarding the involvement either way of parties in each other but instead about submitting earthly things to earthly authorities and spiritual things to The spiritual authority. That is made clearer through other NT passages the clearest examples being 1 Peter 2:13-17.
13 Submit yourselves for the Lord’s sake to every human authority: whether to the emperor, as the supreme authority, 14 or to governors, who are sent by him to punish those who do wrong and to commend those who do right. 15 For it is God’s will that by doing good you should silence the ignorant talk of foolish people. 16 Live as free people, but do not use your freedom as a cover-up for evil; live as God’s slaves. 17 Show proper respect to everyone, love the family of believers, fear God, honor the emperor.
And Romans 13:1-7 which says the same while also including a similar admonition as the passage in question.
7 Give to everyone what you owe them: If you owe taxes, pay taxes; if revenue, then revenue; if respect, then respect; if honor, then honor.
There are exceptions especially when it comes to doing God's will. Joshua 2 shows Rahab's disregard of the king to hand over Israel's two spies. In Acts we see this a few times.
18 Then they called them in again and commanded them not to speak or teach at all in the name of Jesus. 19 But Peter and John replied, “Which is right in God’s eyes: to listen to you, or to him? You be the judges! 20 As for us, we cannot help speaking about what we have seen and heard.”
Acts 5:29 Peter and the other apostles replied: “We must obey God rather than human beings!