Paul cites an example of the kind of conduct that was either directly part of halakha or what it produced. It connects to Peter's experience in Acts 10:28:
And [Peter] said unto them, Ye know how that it is an unlawful thing for a man that is a Jew to keep company, or come unto one of another nation; but God hath shewed me that I should not call any man common or unclean.
The first part of verse 28 has a direct tie to halakha. God had given Peter a vision, recorded earlier in the chapter, in order to instruct him that his perception, his interpretation, was wrong. He was not supposed to call any man common or unclean simply because he had been born to some other racial group or ethnic family other than Jewish.
God's law commanded Israelites to do no such thing as refuse to eat with the Gentiles or even keep company with them. This is a practice derived from Judaism. Even though Peter knew this, he still became carried away into gross hypocrisy when the conditions were right, thus giving us an opportunity to learn that, when Paul is condemning law in the book of Galatians, he is not condemning God's law, but laws men added, thinking they were doing God service.
Here is what happened. Peter came to Antioch for some unstated reason. The church in the town of Antioch was predominately a Gentile church, and while he was there, he circulated freely with the Gentiles. A bit later, though, some Jews arrived, claiming they were from James. Their presence, and possibly their arguments, influenced Peter to withdraw from the Gentiles. So strong was this influence that even Barnabas, Paul's traveling companion on so many of his journeys, was affected so that he withdrew too.
What these Jews—and the apostles caught in it—were doing was effectively driving the church apart! Their teachings and actions were erecting a wall between Jew and Gentile. They were influencing Jews to think they were better than Gentiles, and the Gentiles, that they were inferior unless they submitted to the Jews' standard. The Gentiles wanted to do the right thing, and in their childish ignorance, they began to be led astray. All this was dividing the church.
The standard these Jews taught came neither from God's law nor from the gospel, and the fruit it was producing was class distinction and respect of persons. It came from halakha, part of the Oral Law that frequently had nothing in harmony with God's law.
John W. Ritenbaugh
The Covenants, Grace, and Law (Part 25)