Peter, not Paul is the means of opening the door of the kingdom to the Gentiles. In this way the unity of the assembly and of the work of all the Apostles is maintained. Peter has to learn that God is breaking down the wall between Jew and Gentile. Cornelius has to learn that he needed to be saved.
V.1-8 Here is a very devoted man. But, though so God-fearing, so kind, so generous, he needed to be saved (11:14). Many people today are fine-living people, but they do not want Christ.
V.9-18 Peter, a good living Jew, was proud of law-keeping, not realizing that God was now reaching out in blessing to the Gentiles too. As a Jew, he would not eat any unclean animal. God had given very explicit instruction about this in Leviticus 11 and Deuteronomy 14. And he thought Gentiles were unclean too. We too can cling to traditions which are not according to the Scriptures.