Why was Peter still under the law and covenant promises after the Messiah came?
Peter remained under the law and covenant promises because he came from that Jewish background; nothing had changed except Israel's Messiah had appeared and presented Himself as King, Redeemer, and Messiah, yet the nation largely rejected Him. Peter kept appealing to Israel to repent so He could return as their King.
Les explains that Peter's perspective was shaped by his Jewish background. He was still under the covenant promises and the law, and nothing had changed except that their Messiah had made His appearance. Messiah had presented Himself to Israel as their King, Redeemer, and Messiah, but the nation for the most part rejected Him and crucified Him. In Acts, Peter appeals to Israel over and over to repent of crucifying their Messiah, proclaiming that God had raised Him from the dead and that He was alive and still in a position to come back and be their King. Israel, however, went on in unbelief until Acts chapter 9, when God saved Saul of Tarsus and announced He would send him far hence to the Gentiles. That is when things began to change, leading to the statement in Romans chapter 6 that we are not under law but under grace. Judaizers from Jerusalem then tried to undermine the Gentile converts, insisting they must keep the Law of Moses and be circumcised.