Why did Paul refuse to give in to the Judaizers?
Paul refused because surrendering to the Judaizers' demand for Gentile circumcision and law-keeping would have removed persecution but ended the offense of the cross. He stood firm so the truth of the Gospel would continue with the Gentiles, rather than let the Church die and Satan gloat.
Les explains that Paul says if he had preached what the Judaizers from Jerusalem preached, he would have had clear sailing, no opposition, and would not be under persecution. Their main reason for hating and fighting him was that he would not give in to circumcising these Gentile believers. If he stopped preaching against legalism and the Law, then the offense of the cross would be brought to an end. All Paul would have had to do was succumb to the pressure, give in, and say he would preach circumcision and law keeping so they would get off his back. God would not have been satisfied, but the world would have been, and the Church would have died a natural death while Satan gloated into eternity. But Paul did not do that. He never backed off and never gave in to the Judaizers, and as seen in Galatians chapter 2, he stood his ground so that the truth of the Gospel might continue with the Gentiles. He suffered persecution every day of his life, and later it came under the heavy hand of the Roman government.