That said, however, he feels that his comments have been taken out of context. At a press conference in Sharonville, Ohio, Tuesday, Carson said, "I don't care what a person's religious beliefs are or what their religious heritage is. If they embrace American culture, if they embrace our Constitution and are willing to place that above their religious beliefs, I have no problem with that. I said that. Why is it so impossible for people to hear that?"
He said earlier, in an interview with Fox News, that his view does not just apply to Muslims.
"If, for instance, you believe in a theocracy, I don't care if you're a Christian. If you're a Christian and you're running for president and you want to make this into a theocracy, I'm not going to support you. I'm not going to advocate you being the president."
He also said that a Muslims rejecting their religion and swearing to uphold the Constitution, "would be considered infidels and heretics, but at least then I will be quite willing to support them."
Carson's initial comments that he would "not advocate that we put a Muslim in charge of this nation" came when "Meet the Press" moderator Chuck Todd asked him about fellow GOP candidate Donald Trump's silence on President Obama's faith, after a questioner at an event stated that Mr. Obama was a Muslim and also not an American.
Asked whether a person's faith should matter, Carson said, "I guess it depends on what that faith is. If it's inconsistent with the values and principles of America, then of course it should matter. But if it fits within the realm of America and consistent with the Constitution, no problem."
The Associated Press reported that Carson gained 100,000 new Facebook friends after his comments.
"People in Iowa particularly, are like, 'Yeah! We're not going to vote for a Muslim either," his campaign manager, Barry Bennett, told the AP. "I don't mind the hubbub. It's not hurting us, that's for sure."
The executive director of the Council on American-Islamic relations said he should drop out of the race, and many of Carson's fellow candidates have denounced his statements. Former Hewlett Packard CEO Carly Fiorina said, "I think that's just wrong." Sen. Lindsey Graham, R-South Carolina, said the remarks revealed Carson was not ready to be commander in chief.
"The Constitution specifies there shall be no religious test for public office and I am a Constitutionalist," Sen. Ted Cruz, R-Texas, said on Iowa public television in response.
Even Trump - who let a commenter from the audience call President Obama a Muslim without challenging him - and has no regrets about it - indicated he would likely be open to someone of that faith in the White House.
"If properly vetted - the proper people properly vetted - going through an election, I think that anybody that is able to win an election will be absolutely fine," he said on Fox News Monday night.
There is currently no one running as a Democrat or Republican with at least one percent support who is a Muslim.
Katiana Krawchenko contributed to this report