Instead, he's doubling down on a formal objection that Islam is incompatible with Western civilization -- and the Constitution -- unless Muslims become extremely flexible on its law. America should be a "Judeo-Christian" nation that defends the rights of women.
The Anti-Defamation League posits an objection based on, I gather, concerns about how we might receive presidential candidates who came out against Jews the way Carson is coming out against (at least fundamentalist) Muslims. On the other hand, Front Page Magazine asks whether we can really expect "a black man to prove his tolerance by endorsing a religion that practices slavery?" Given the currency of the rise of ISIS and its practice of slavery justified by appeals to sha'riah law, that's a reasonable question. But they aren't just talking about ISIS: Front Page cites a Human Rights Watch report based on compensation levels for death in Jeddah based on the free or unfree status of the dead.
I'm not sure that 'protection of the rights of women' is even a Judeo-Christian thing, but rather a specifically Medieval European tradition. It has a Judeo-Christian root in the idea, expressed by Aquinas among others, that God created both male and female and loves them both equally. But it is a chivalric ideal to defend the rights of women as an especial duty of good men. Malory put it in the Arthurian oath, taken each year at the feast of Pentecost:
“The king stablished all his knights, and gave them that were of lands not rich, he gave them lands, and charged them never to do outrageousity nor murder, and always to flee treason; also, by no mean to be cruel, but to give mercy unto him that asketh mercy, upon pain of forfeiture of their worship and lordship of King Arthur for evermore; and always to do ladies, damosels, and gentlewomen succor upon pain of death. Also, that no man take no battles in a wrongful quarrel for no law, ne for no world’s goods. Unto this were all the knights sworn of the Table Round, both old and young. And every year were they sworn at the high feast of Pentecost.” (Le Morte d'Arthur, pp 115-116)Emphasis added, but it is already quite emphasized in the text: on pain of death. The oath states that Arthur would have hung one of his knights if he failed to defend a lady who needed his help.