St Paul never converted to Christianity.
For one, at the time of the event on the road to Damascus, there was no such thing as "Christianity". There were multiple threads of Judaism, including Pharisees, Sadducees, Essenes, and traditions within these sects such as zealotry. St Paul was a self-described zealot of the school of the Pharisees. There was no "religion" called Judaism - the various Judaisms were the customs, life, and religious practices of the people of Judaea, which is where it gets the name "Jew" from. There were also non-Jewish "god-fearers" around the edges of Jewish belief - gentiles who were interested in Judaism or psuedo-Jewish philosophy, but were not themselves Jewish.
Similarly, there was no "Roman religion" or "Greek religion". There were the ways of life that Romans and the various Greeks practiced - which included religious, secular, and cultural law and practices. The Lex of a Roman was the same as the Torah of a Judaean was the same as a Nomos of a Greek - your way of life. Your nomos, law, lex, torah was what made you a Roman, or Jew, or Greek - not your nature, genetics, or genealogy. If you were a Judaean you followed Torah; following Torah is what made you Judaean. If you did not follow Torah you were not a Judaean any longer. (Maccabees is a great study on this!)
St Paul - and all the other Apostles - never ceased following the Torah. He never stopped being a Judaean, never stopped celebrating the religious customs he practiced. He continued to worship in the Temple, continued to go to the Synagogue. He continued being a Pharisee. He perhaps did give up his zealotry or it at least was baptized into more productive efforts than violence.
St Paul also never changed his name. He was from Tarsus, a city which had won free and independent status from Julius Caesar because of their support for him. St Paul's father likely became a citizen at that time, and St Paul inherited his Roman citizenship accordingly. He grew up around non-Jews and likely interacted with them regularly. He also had clear knowledge of non-Jewish philosophers and writers. It was common at the time for Hebrew people in situations like this to have two names - in the scriptures we have John called Mark, Silas / Silvanas, Joseph / Justus, Simeon / Niger, Thomas / Didymus...and Saul called Paul. The Lord refers to him by his Hebrew name "Saul" and never changes his name to Paul. In the Acts narrative he goes from being Saul or Saul called Paul to just Paul when his mission status takes him into the area of the non-Jewish world.
So what did happen?
St Paul says he continued being a Pharisee, and continued in the faith of his fathers. In his most detailed account in Acts 26 he says lived strictly as a Pharisee and preaches nothing more than the promise made by God to his fathers, "saying nothing but what the prophets and Moses said would come to pass."
Instead of conversion, when St Paul talks about what happened to him, he describes it as a revelation and calling - he explicitly likens himself to the prophets of the Old Testament.
For example:
I, the LORD, have called you for a righteous purpose (Isaiah)
I will make you as a light for the nations, that my salvation may reach to the end of the earth (Isaiah)
Before I formed you in the womb I knew you, and before you were born I set you apart and appointed you as a prophet to the nations (Jeremiah)
St Paul
But the Lord said to him, "Go, for he is a chosen instrument of mine to carry my name before the Gentiles and kings and the children of Israel..." (Acts 9:15)
For so the Lord has commanded us, saying, "I have made you a light for the Gentiles, that you may bring salvation to the ends of the earth." (Acts 13:47)
Paul, a servant of Christ Jesus, called to be an apostle, set apart for the gospel of God (Romans 1:1)
But when he who had set me apart before I was born, and who called me by his grace, was pleased to reveal his Son to me, in order that I might preach him among the Gentiles (Gal 1:15-16)
What happened on the road to Damascus was a few things:
1) St Paul correctly understood that the Messiah he was waiting for had already come, and was Jesus of Nazareth.
2) St Paul's timeline shifted accordingly: no longer was he waiting in expectation for the Messiah, or trying to cleans the land through his Pharisaic zeal to bring about the Messiah's coming... but instead he realized he was in the Messianic age!
3) St Paul was called by Jesus through multiple revelations that through the gospel to the gentiles the prophecies of the Old Testament beginning in Genesis
And this is St Paul's gospel to the nations.
-In the past the people of the nations were separated from the God of Israel and were ignorant of Him, living in sin. He tolerated their sin in mercy until now, but the time of his mercy was ending.
-That the Messiah died and rose again, and in this one-time atonement cleansed not just the Jewish temple and people but the whole world and all people, opening the way for all who were faithful to him to be pleasing to God.
-That Abraham was the father of many nations, and was found faithful and pleasing to God before he was circumcised. So if they were faithful to God, they too could be found pleasing to God in their uncircumcision, and have Abraham as their father: they did not need to be Jewish or be circumcised to approach God.
-The Messiah is returning and will judge the living and the dead for what they have done.
-To repent, therefore, be faithful to the Messiah, and stop being a Roman or Greek or Scythian with those ways of life that were not compatible with faith in Jesus, and begin being a new creation by following the teaching of Jesus which fulfills the Torah
-And that that when the northern kingdom (known in the scriptures as Ephraim cf Ezek 37:16) was scattered into the nations the seeds were planted; now that the people of the nations are coming to faithfulness in Messiah, Ephraim brings in the fullness of the gentiles, and simultaneously restores the lost tribes of Israel. The faithful non-Jews coming to Jesus become part of All Israel, and have Abraham and the Patriarchs as their fathers.
None of this is Christian against the Pharisaic Judaism St Paul already practiced and continued to practice for the rest of his life.
For one, at the time of the event on the road to Damascus, there was no such thing as "Christianity". There were multiple threads of Judaism, including Pharisees, Sadducees, Essenes, and traditions within these sects such as zealotry. St Paul was a self-described zealot of the school of the Pharisees. There was no "religion" called Judaism - the various Judaisms were the customs, life, and religious practices of the people of Judaea, which is where it gets the name "Jew" from. There were also non-Jewish "god-fearers" around the edges of Jewish belief - gentiles who were interested in Judaism or psuedo-Jewish philosophy, but were not themselves Jewish.
Similarly, there was no "Roman religion" or "Greek religion". There were the ways of life that Romans and the various Greeks practiced - which included religious, secular, and cultural law and practices. The Lex of a Roman was the same as the Torah of a Judaean was the same as a Nomos of a Greek - your way of life. Your nomos, law, lex, torah was what made you a Roman, or Jew, or Greek - not your nature, genetics, or genealogy. If you were a Judaean you followed Torah; following Torah is what made you Judaean. If you did not follow Torah you were not a Judaean any longer. (Maccabees is a great study on this!)
St Paul - and all the other Apostles - never ceased following the Torah. He never stopped being a Judaean, never stopped celebrating the religious customs he practiced. He continued to worship in the Temple, continued to go to the Synagogue. He continued being a Pharisee. He perhaps did give up his zealotry or it at least was baptized into more productive efforts than violence.
St Paul also never changed his name. He was from Tarsus, a city which had won free and independent status from Julius Caesar because of their support for him. St Paul's father likely became a citizen at that time, and St Paul inherited his Roman citizenship accordingly. He grew up around non-Jews and likely interacted with them regularly. He also had clear knowledge of non-Jewish philosophers and writers. It was common at the time for Hebrew people in situations like this to have two names - in the scriptures we have John called Mark, Silas / Silvanas, Joseph / Justus, Simeon / Niger, Thomas / Didymus...and Saul called Paul. The Lord refers to him by his Hebrew name "Saul" and never changes his name to Paul. In the Acts narrative he goes from being Saul or Saul called Paul to just Paul when his mission status takes him into the area of the non-Jewish world.
So what did happen?
St Paul says he continued being a Pharisee, and continued in the faith of his fathers. In his most detailed account in Acts 26 he says lived strictly as a Pharisee and preaches nothing more than the promise made by God to his fathers, "saying nothing but what the prophets and Moses said would come to pass."
Instead of conversion, when St Paul talks about what happened to him, he describes it as a revelation and calling - he explicitly likens himself to the prophets of the Old Testament.
For example:
I, the LORD, have called you for a righteous purpose (Isaiah)
I will make you as a light for the nations, that my salvation may reach to the end of the earth (Isaiah)
Before I formed you in the womb I knew you, and before you were born I set you apart and appointed you as a prophet to the nations (Jeremiah)
St Paul
But the Lord said to him, "Go, for he is a chosen instrument of mine to carry my name before the Gentiles and kings and the children of Israel..." (Acts 9:15)
For so the Lord has commanded us, saying, "I have made you a light for the Gentiles, that you may bring salvation to the ends of the earth." (Acts 13:47)
Paul, a servant of Christ Jesus, called to be an apostle, set apart for the gospel of God (Romans 1:1)
But when he who had set me apart before I was born, and who called me by his grace, was pleased to reveal his Son to me, in order that I might preach him among the Gentiles (Gal 1:15-16)
What happened on the road to Damascus was a few things:
1) St Paul correctly understood that the Messiah he was waiting for had already come, and was Jesus of Nazareth.
2) St Paul's timeline shifted accordingly: no longer was he waiting in expectation for the Messiah, or trying to cleans the land through his Pharisaic zeal to bring about the Messiah's coming... but instead he realized he was in the Messianic age!
3) St Paul was called by Jesus through multiple revelations that through the gospel to the gentiles the prophecies of the Old Testament beginning in Genesis
- That through Abraham's seed all the nations of the earth would be blessed
- That through Joseph's son Ephraim would be the fullness of the gentiles
- That all the lost tribes of Israel would be restored
- That in the Messiah the nations would come to faithful worship of Yahweh the God of Israel
And this is St Paul's gospel to the nations.
-In the past the people of the nations were separated from the God of Israel and were ignorant of Him, living in sin. He tolerated their sin in mercy until now, but the time of his mercy was ending.
-That the Messiah died and rose again, and in this one-time atonement cleansed not just the Jewish temple and people but the whole world and all people, opening the way for all who were faithful to him to be pleasing to God.
-That Abraham was the father of many nations, and was found faithful and pleasing to God before he was circumcised. So if they were faithful to God, they too could be found pleasing to God in their uncircumcision, and have Abraham as their father: they did not need to be Jewish or be circumcised to approach God.
-The Messiah is returning and will judge the living and the dead for what they have done.
-To repent, therefore, be faithful to the Messiah, and stop being a Roman or Greek or Scythian with those ways of life that were not compatible with faith in Jesus, and begin being a new creation by following the teaching of Jesus which fulfills the Torah
-And that that when the northern kingdom (known in the scriptures as Ephraim cf Ezek 37:16) was scattered into the nations the seeds were planted; now that the people of the nations are coming to faithfulness in Messiah, Ephraim brings in the fullness of the gentiles, and simultaneously restores the lost tribes of Israel. The faithful non-Jews coming to Jesus become part of All Israel, and have Abraham and the Patriarchs as their fathers.
None of this is Christian against the Pharisaic Judaism St Paul already practiced and continued to practice for the rest of his life.