How did Peter publicly respond to God saving the Gentiles apart from Israel?
Peter did not publicly announce it to Israel. After Cornelius' household, he and those with him remained silent among the children of Israel for another eight or ten years, never casting a word that God was now dealing with Gentiles on His own ground apart from the nation of Israel.
Les explains that the sign at Cornelius' house was particularly meant to convince Jews, and those who went with Peter should have returned to Jerusalem spreading the word that God was now ready to turn to the Gentiles without Israel. But they did not. When you go back to Acts chapter 15, Peter and those who evidently went with him had never said a word for another eight or ten years. Not one word did they cast among the children of Israel that God was now dealing with the Gentiles on His own ground and not on the basis of using the nation of Israel. So Peter's public response, in this context, was silence rather than proclamation.
More Questions
Did Peter ever tell Israel that God was accepting Gentiles apart from the nation of Israel after Cornelius?
Peter and the Jewish believers who evidently went with him never said a word for eight or ten years, and never told Israel that God was now dealing with the Gentiles directly rather than through the nation of Israel.