Carson clarifies

A visit to the dentist in medieval Persia. Via medievalists.net.

What's wrong with Muslims?

"Under Islamic Law, homosexuals – men and women alike – must be killed. Women must be subservient. And people following other religions must be killed," he continued. "I know that there are many peaceful Muslims who do not adhere to these beliefs. But until these tenants (sic) are fully renounced…I cannot advocate any Muslim candidate for President."

Obviously that's a lot worse than Biblical literalist Christianity in which only male homosexuals need to be put to death and the girls get away scot-free. I know that there are many peaceful Christians who do not agree with the death penalty for sodomy, including the members of Dr. Carson's Seventh-Day Adventist Church, who hold that

Not only those with homosexual tendencies, butt [sic] all persons who are trapped in behaviors or relationships that cause anxiety, shame, and guilt need the sympathetic ear of a trained and experienced Christian counselor.  No behavior is beyond the reach of God’s healing grace”

which certainly seems much nicer, even though Adventists do insist that people have to stop doing that stuff. Their position is certainly more inflexible, though, than that of the Hanafi jurists (the largest of the four schools of Sunni jurisprudence), who hold that homosexuality is not a hadd crime (for which punishment is specified in the Qur'an) and whether it needs to be punished at all is a matter for individual judges to decide and subject to changing social conventions:

Unlike in Christianity, there was a time in the Middle Ages when same-sex relations among Muslims were very broadly accepted socially and by many theologians as well:

The four schools of shari'a (Islamic law) disagreed on what punishment is appropriate for liwat. Abu Bakr Al-Jassas (d. 981 AD/370 AH) argued that the two hadiths on killing homosexuals "are not reliable by any means and no legal punishment can be prescribed based on them",^[23] and the Hanafi school held that it does not merit any physical punishment, on the basis of a hadith that "Muslim blood can only be spilled for adultery, apostasy and homicide"; against this the Hanbali school inferred that sodomy is a form of adultery and must incur the same penalty, i.e. death.

Why the harsh view appears to have become more dominant in modern times I don't know, but perhaps they were influenced by contact with European Christians in the age of imperialism.

As Dr. Carson doesn't cite his sources for his interpretation of Islamic law, I can't evaluate it, but I find that the Prophet didn't feel that way about "people following other religions" with whom he wasn't at war, as governor of Medina:

“Beware!  Whoever is cruel and hard on a non-Muslim minority, or curtails their rights, or burdens them with more than they can bear, or takes anything from them against their free will; I (Prophet Muhammad) will complain against the person on the Day of Judgment.” (Abu Dawud)

Carson on the Hannity show went on to say:

"Now, if someone has a Muslim background, and they’re willing to reject those tenets and to accept the way of life that we have, and clearly will swear to place our Constitution above their religion, then of course they will be considered infidels and heretics, but at least I would then be quite willing to support them."

He's allowed to vote that way if he wants, but he's very wrong on what the Constitution demands; I'd rather look at the candidate not through the lens of my beliefs about her or his religion but how she or he has lived and spoken. I'd rather follow the advice of John F. Kennedy, September 12 1960:

I ask you tonight to follow in that tradition, to judge me on the basis of my record of 14 years in Congress, on my declared stands against an ambassador to the Vatican, against unconstitutional aid to parochial schools, and against any boycott of the public schools (which I have attended myself)— instead of judging me on the basis of these pamphlets and publications we all have seen that carefully select quotations out of context from the statements of Catholic church leaders, usually in other countries, frequently in other centuries, and always omitting, of course, the statement of the American Bishops in 1948, which strongly endorsed church-state separation, and which more nearly reflects the views of almost every American Catholic.