American society allows the freedom of religion, so being a Muslim does not nullify being an American. A more apt analogy to religious apostasy would be treason, which many societies have in the past punished by execution, including America. So perhaps you should ask yourself if American society would be justified in executing Muslim converts when the Constitution gives them that freedom. I say no, that Americans would not be justified in infringing upon the religious rights that their laws provide. As for what another country may do with apostates or traitors, then that is based upon their laws.
This may shock you, but the Islamic punishment for apostasy is not simply execution and they are not always executed. There is a due process and the punishment is depending upon the case. If the apostate were to work against the Muslim State, allying, aiding and abetting its enemies, then this person would be guilty of treason, just as they would be in America and receive a similar punishment. In some cases, the apostate is not executed and allowed to remain within the society.