The Jews need to believe and obey the Gospel of Jesus Christ too.
Paul opposes Peter's hypocrosy in Galatians 2:11-21
Peter defends Paul and says they are saved by Grace just like the gentiles - Acts 15
7 And after there had been much debate, Peter stood up and said to them, “Brothers, you know that in the early days God made a choice among you, that by my mouth the Gentiles should hear the word of the gospel and believe. 8 And God, who knows the heart, bore witness to them, by giving them the Holy Spirit just as he did to us, 9 and he made no distinction between us and them, having cleansed their hearts by faith. 10 Now, therefore, why are you putting God to the test by placing a yoke on the neck of the disciples that neither our fathers nor we have been able to bear? 11 But we believe that we will be saved through the grace of the Lord Jesus, just as they will.”
2 Peter 3:15-16 - Peter calls Pauls writings scripture
15 And count the patience of our Lord as salvation, just as our beloved brother Paul also wrote to you according to the wisdom given him, 16 as he does in all his letters when he speaks in them of these matters. There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures
21 But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— 22 the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: 23 for all have sinned and fall short of the glory of God, 24 and are justified by his grace as a gift, through the redemption that is in Christ Jesus, 25 whom God put forward as a propitiation by his blood, to be received by faith. This was to show God’s righteousness, because in his divine forbearance he had passed over former sins. 26 It was to show his righteousness at the present time, so that he might be just and the justifier of the one who has faith in Jesus.
3 Now there was a man of the Pharisees named Nicodemus, a ruler of the Jews. 2 This man came to Jesus by night and said to him, “Rabbi, we know that you are a teacher come from God, for no one can do these signs that you do unless God is with him.” 3 Jesus answered him, “Truly, truly, I say to you, unless one is born again he cannot see the kingdom of God.” 4 Nicodemus said to him, “How can a man be born when he is old? Can he enter a second time into his mother's womb and be born?” 5 Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God. 6 That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. 7 Do not marvel that I said to you, ‘You must be born again.’ 8 The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.”
9 Nicodemus said to him, “How can these things be?” 10 Jesus answered him, “Are you the teacher of Israel and yet you do not understand these things? 11 Truly, truly, I say to you, we speak of what we know, and bear witness to what we have seen, but you do not receive our testimony. 12 If I have told you earthly things and you do not believe, how can you believe if I tell you heavenly things? 13 No one has ascended into heaven except he who descended from heaven, the Son of Man. 14 And as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up, 15 that whoever believes in him may have eternal life.
16 “For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life. 17 For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him. 18 Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God. 19 And this is the judgment: the light has come into the world, and people loved the darkness rather than the light because their works were evil. 20 For everyone who does wicked things hates the light and does not come to the light, lest his works should be exposed. 21 But whoever does what is true comes to the light, so that it may be clearly seen that his works have been carried out in God.”