Gal 2:14 But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?
Gal 2:15 We who are Jews by nature, and not sinners of the Gentiles,
Gal 2:16 Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justified.
You have omitted the most important part of this story. Here, let me add the rest, so we can more clearly understand what is happening.
12 For before that certain came from James, he did eat with the Gentiles: but when they (Pharisees) were come, he withdrew and separated himself, fearing them which were "of the circumcision". (Pharisees religion)
13 And the other Jews dissembled likewise with him; insomuch that Barnabas also was carried away with their dissimulation.
It was a commandment of men the Pharisees taught for doctrine, not to eat with or keep company with a Non-Jew.
It is important to understand these truths. Peter had repented from living in the commandments and traditions of the disobedient Jews, but was turning again, as Paul teaches, "
But now, after that ye have known God, or rather are known of God, how turn ye again to the weak and beggarly elements, whereunto ye desire again to be in bondage?
Peter was reverting back to the commandments and Philosophies of men the Pharisees taught for doctrines.
Acts 10:
28 And he 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.
This is one of many "Commandments of men" the Pharisees taught for doctrines. It was never a commandment from God not to eat dinner or sit with a non-Jew that was sojourning with them.
Both the Jew and the Gentile needs forgiveness/justification. The Jews were still promoting their twisted version of the Levitical Priesthood sacrificial "works of the law" for Justification. They had made a business out of it, selling salvation on the Sabbath Days. Jesus actually whipped men for doing this. Paul is telling them what Moses and the Gospel of Christ has promoted since the Exodus.
Deut 10:
16 Circumcise therefore the foreskin of your heart, and be no more stiffnecked. 17 For the LORD your God is God of gods, and Lord of lords, a great God, a mighty, and a terrible, which regardeth not persons, nor taketh reward: 18 He doth execute the judgment of the fatherless and widow, and loveth the stranger, in giving him food and raiment. 19 Love ye therefore the stranger: for ye were strangers in the land of Egypt.
The deceiver uses some of Paul's words to trick men into living in disobedience to God's instruction in righteousness, like the other voice in the garden, who quoted some of God's Word, did to Eve. But Paul knew better, and he and I "press towards the High calling of God" (Be ye perfect) which was in Christ Jesus. Both of us Laboring as Paul teaches.
2 Cor. 5:
9 Wherefore we labour, that, whether present or absent, "we may be accepted of him". 10 For we must all appear before the judgment seat of Christ; that every one may receive the things done in his body, according to that he hath done, whether it be good or bad.
It is unprofitable to pick one paragraph from Paul's letters, without considering all of his teaching.
Notice.
1. Peter lived as a Gentile lived.
2. Peter considered himself justified in such actions.
3. Notice the connection to the moral demands of the law they claimed (Peter included).
4. Notice the TRUTH. There is no justification either morally in the law (Torah) or anything other than belief in Jesus Christ.
There is no Law given that can forgive sins. There is no forgiveness without repentance and turning away from all your transgressions. And the "Doers" of God's Law are justified, not the hearers only. Not by the temporary sacrificial "works of the Law" put in place "Till the Seed should come" that the Pharisees were still promoting. But by Faith, that if I turn from my ways, and turn to God in obedience, that God will give me to Jesus for saving.
Even satan believes in Jesus, so clearly there is more to salvation than calling Jesus Lord, Lord.
The life of Peter itself shows just how impactful Judaism was to the TRUTH of Jesus Christ.
This is why I asked, and have not been answered yet, what is Judaism? Is it walking in the commandments, Judgments and Statutes of God, like Jesus and EVERY Example of Faithful man in the entire bible? Or is it walking in this world's religious philosophies, traditions, high days and doctrines of men?
Why won't religious men answer this simple question?
It is dangerous. In my view, it is one of the most dangerous false doctrines in the history of Christianity.
Which definition of Judaism is dangerous? The Laws, doctrines and instruction in righteousness from God that Jesus walked in? Or the other voice in the garden, who quotes some of God's Word to trick men into rejecting God's Judgments and Commandments, like the rebellious Jews did?
It's an important question that must be answered for honest discussion of this topic, in my view.
So much so that it played out in the very lives of almost all of the apostles.
Witnessing this issue with Peter should be enough for you to be concerned yourself.
I agree that if Judaism is the religious philosophies, traditions, commandments and doctrines of men the Pharisees laid on the backs of men who listened to them, then it is most dangerous, as Peter shows when he "turned again" to his past life, "
Wherein in time past he walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience".
But if Judaism is "Yielding myself" a servant to obey God, as Jesus did, walking in/Living by all the Commandments and Judgments of God, as Jesus did, and "yielding" my mind and body as "instruments of Righteousness" unto God, also as Jesus did. Then it makes perfect sense why Jesus warned, not of Atheists, or Islam or Buddhist, but specifically of religious men who call Him Lord, Lord, who "Come in His Name" but are not "doers" of His sayings, rather, men who walk in the doctrines, philosophies, high days and judgments of the religious system of this world.