No. Nobody gives him a name in Acts 9, and Luke is still calling him Saul four chapters later. The verse everybody points to does not report a change at all.
The Damascus road hands out no names
And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me?
Acts 9:4 (KJV)
The voice says Saul, twice. Ananias, sent to him three days later, greets him as Brother Saul. He is baptized in verse eighteen and he is still Saul in verse nineteen, still Saul when he starts preaching, still Saul when the disciples in Jerusalem are afraid of him.
The closest thing to a commission in that chapter is the Lord telling Ananias that this man is a chosen vessel to bear my name before the Gentiles, and the name he is handed belongs to somebody else.
Luke keeps writing Saul for years of narrative
Go through Acts and count. Saul appears in twenty-two verses across eight chapters, not counting the one place where the Saul in question is the king. Paul appears in a hundred and twenty-nine verses across sixteen. Luke writes both names in only three chapters, and in two of those Saul survives solely inside a quoted voice.
Between the conversion and that chapter, Luke writes Saul after the baptism, after the first sermons, after Barnabas goes to Tarsus to fetch him, after the famine relief run to Jerusalem, and after the Holy Spirit says separate me Barnabas and Saul for the work. That last one is the commissioning for the missionary journeys, where the Spirit itself uses the old name.
Now the verse everybody cites
Then Saul, (who also is called Paul,) filled with the Holy Ghost, set his eyes on him,
Acts 13:9 (KJV)
Who also is called Paul. Not who was renamed, not who became, not who took the name. The sentence tells you something that was already true.
And there is a detail underneath it. In the Greek there is no verb. The phrase is an article and the word for also, two words, with both names in the same case. The King James supplies is called and prints it in italics to say so. Whatever that construction means, it is the ordinary formula for a man who has two names, and it does not contain an act of naming.
Paulus enters the book two verses early
Look at who is standing there.
Which was with the deputy of the country, Sergius Paulus, a prudent man; who called for Barnabas and Saul, and desired to hear the word of God.
Acts 13:7 (KJV)
Barnabas and Saul, as Luke still writes it, are on Cyprus in front of the Roman proconsul, and Luke gives the proconsul’s name before he gives the apostle’s. In Greek the proconsul’s name and the apostle’s name carry the same number in a concordance. They are the same name.
The word Paulus therefore enters the book of Acts two verses before it is first used of Saul, attached to a Roman official, in the first scene where Luke shows the apostle working on Roman ground in front of a Roman audience.
Luke never narrates him as Saul again after that verse, and the name Paul carries the hundred and twenty-eight verses that follow it.
None of that means Paul took the name from the proconsul. A man born a Roman citizen carried a Roman name from birth, and this one was born a citizen. A Roman officer tells him what his own citizenship cost, and Paul answers, But I was free born, in Acts 22:28.
He tells the story twice, decades later, and the voice still says Saul
In Acts 22 and Acts 26, standing in chains, he retells the Damascus road to two different audiences. Both times the voice says Saul, Saul, and in the second telling he adds a detail.
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.
Acts 26:14 (KJV)
The man now going by Paul, decades on, says that the name in that moment was his Hebrew one, spoken in Hebrew, and neither retelling contains a renaming. The voice gives him the same instruction both times: get up and go.
The Greek keeps the two forms apart to the letter. Five places in Acts have the name spoken to him, 9:4, 9:17, 22:7, 22:13 and 26:14, and all five use the indeclinable Hebrew form. Luke’s own narration uses the Greek form, and a concordance gives the two different numbers.
In his own letters he never writes Saul once
Thirteen letters, and the name does not appear in any of them, in any context, while he opens every one of them as Paul.
The place you would expect it is Philippians 3, where he lists what he used to be. Circumcised the eighth day, of the stock of Israel, of the tribe of Benjamin, a Hebrew of the Hebrews, a Pharisee, a persecutor of the church. A full recital of the old life, and the old name is not in it.
Which is interesting, because he does keep the tribe. King Saul was a Benjamite, and Paul says twice that he is one too, in Romans 11 and in that Philippians list, so the connection he might have dropped is the one he holds on to.
The name Saul is in his mouth in four places in the whole book. Three are quotation, the voice on the road twice and Ananias once, and the fourth is his own. Twelve verses after Luke stopped calling him Saul, Paul is preaching in the synagogue at Antioch in Pisidia, and tells them that God gave Israel Saul the son of Cis, a man of the tribe of Benjamin.
A clean before and after
The renaming story is popular because it is a good shape: old man, old name, new man, new name, all at the moment of the light. Tradition built the same shape for Mary Magdalene, out of the same wish for a story that turns on a single afternoon. The text here gives a slower and less convenient one. A man keeps his name through the blindness and the baptism and the first decade of preaching, and the second name surfaces at the point where his work turns toward Rome, in a sentence that says he had it all along.
Readers finish sentences the writer left open, the way they gave the forbidden fruit a species, and then the finished version is the one everybody remembers. What conversion changed in this man is on every page of Acts. It was not his stationery.