I think a lot of confusion arises with the phrase "Calling on the name of the Lord". I know a lot of people who interpret that as praying the Sinner's Prayer, or asking Jesus into your heart as your personal Savior, etc. If we look at the text, we can find out exactly what it means.
First, Peter tells us in Acts 2:21;36-38
21And it shall come to pass that everyone who calls upon the name of the Lord shall be saved. (Quoting the Prophet Joel)
So how do we call upon the name of the Lord?
36Let all the house of Israel therefore know for certain that God has made him both Lord and Christ, this Jesus whom you crucified."
37Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, "Brothers, what shall we do?" 38And Peter said to them, "Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit.
So first we acknowledge that Jesus is both Lord and Christ, repent and then are baptized for forgiveness of sins, so that we may receive the gift of the Holy Spirit (Salvation).
Then with Paul we find out that not even repenting and praying to God is enough to wash away our sins, or is equivalent to "calling on the name of the Lord".
Acts 9:10-12
10Now there was a disciple at Damascus named Ananias. The Lord said to him in a vision, "Ananias." And he said, "Here I am, Lord." 11And the Lord said to him, "Rise and go to the street called Straight, and at the house of Judas look for a man of Tarsus named Saul, for behold, he is praying, 12and he has seen in a vision a man named Ananias come in and lay his hands on him so that he might regain his sight."
So he had a miraculous experience on the road to Damascus, was struck blind, and is praying to God. Let's continue to Acts 22 for more info on what happened.
Acts 22:12-16
12"And one Ananias, a devout man according to the law, well spoken of by all the Jews who lived there, 13 came to me, and standing by me said to me, 'Brother Saul, receive your sight.' And at that very hour I received my sight and saw him. 14And he said, 'The God of our fathers appointed you to know his will, to see the Righteous One and to hear a voice from his mouth; 15for you will be a witness for him to everyone of what you have seen and heard. 16And now why do you wait? Rise and be baptized and wash away your sins, calling on his name.'
If baptism wasn't necessary for salvation, what else would have saved Paul? He was praying to God, obviously moved by the experience. He fasted for three whole days. He was a devout Jew, well versed in the Law of Moses. He had every other base covered, but Ananias still asked, "And now why do you wait? Rise and be baptized and wash away your sins, calling on his name."