Did Jesus's disciples accept Paul?

No in the scriptures it shows that Paul actually argued with them and detested what they were teaching. Which should bring up question because the original disciples actually took their teachings from Jesus the messenger of God (Greek translation: Christ)

A New Testament perspective:

After Saul's conversion (and some reassurances from a trusted brother), yes, they did.

Acts 9:26-29 - And when Saul had come to Jerusalem, he tried to join the disciples; but they were all afraid of him, and did not believe that he was a disciple. But Barnabas took him and brought him to the apostles. And he declared to them how he had seen the Lord on the road, and that He had spoken to him, and how he had preached boldly at Damascus in the name of Jesus. So he was with them at Jerusalem, coming in and going out. And he spoke boldly in the name of the Lord Jesus and disputed against the Hellenists, but they attempted to kill him.

The Hellenists were not disciples of Christ; they were Greek-speaking Jews born outside of Israel who argued for the Law as opposed to the Gospel of Christ. The only people whose teachings Paul detested were those who opposed Jesus' gospel, and vice-versa. For example:

Acts 13:6-10 - Now when they had gone through the island to Paphos, they found a certain sorcerer, a false prophet, a Jew whose name was Bar-Jesus, who was with the proconsul, Sergius Paulus, an intelligent man. This man called for Barnabas and Saul and sought to hear the word of God. But Elymas the sorcerer (for so his name is translated) withstood them, seeking to turn the proconsul away from the faith. Then Saul, who also is called Paul, filled with the Holy Spirit, looked intently at him and said, "O full of all deceit and all fraud, you son of the devil, you enemy of all righteousness, will you not cease perverting the straight ways of the Lord?"

Furthermore, Paul and Peter both affirm such acceptance in their epistles.

Galatians 2:7-9 - But on the contrary, when they saw that the gospel for the uncircumcised had been committed to me, as the gospel for the circumcised was to Peter (for He who worked effectively in Peter for the apostleship to the circumcised also worked effectively in me toward the Gentiles), and when James, Cephas, and John, who seemed to be pillars, perceived the grace that had been given to me, they gave me and Barnabas the right hand of fellowship, that we should go to the Gentiles and they to the circumcised.

2 Peter 3:14-16 - Therefore, beloved, looking forward to these things, be diligent to be found by Him in peace, without spot and blameless; and consider that the longsuffering of our Lord is salvation - as also our beloved brother Paul, according to the wisdom given to him, has written to you, as also in all his epistles, speaking in them of these things, in which are some things hard to understand, which untaught and unstable people twist to their own destruction, as they do also the rest of the Scriptures.

[Quotes from NKJV]