"Yes, you have to," the retired neurosurgeon and Republican presidential candidate said Sunday in an interview with CNN's Jake Tapper on "State of the Union."
His latest criticism of Islam came the week after he'd said he "would not advocate that we put a Muslim in charge of this nation."
"I would have problems with somebody who embraced all the doctrines associated with Islam," Carson said. "If they are not willing to reject sharia and all the portions of it that are talked about in the Quran -- if they are not willing to reject that, and subject that to American values and the Constitution, then of course, I would."
He argued, though, that the controversy surrounding his comments last week has been overblown.
"Is it possible that maybe the media thinks it's a bigger deal than the American people do?" Carson said. "Because American people, the majority of them, agree and they understand exactly what I am saying."
Carson insisted Sunday that "of course Muslims can be patriots," though his previous comments appeared to presume that Muslims are more devoted to their faith than the U.S. Constitution.
"I've worked with Muslims. I've trained Muslims. I've operated on Muslims. There are a lot of Muslims who are very patriotic. Good Americans and they gladly admit, at least privately, that they don't accept sharia or the doctrines and they understand that Islam is a system of living and it includes the way that you relate to the government," he said.
"And you cannot, unless you specifically, deny that portion of Islam be a Muslim in good standing. Now if that is the case, if you are not willing to reject that, then how in the world can you possibly be the president of the United States."
After several minutes of back-and-forth over his position on Muslims' qualifications for political office, Carson's aide stopped the interview.
Tapper said: "You're assuming that Muslim Americans put their religion ahead of the country."
And Carson responded: "I'm assuming that if you accept all the tenets of Islam that you would have a very difficult time abiding under the Constitution of the United States."
Armstrong Williams, Carson's campaign business manager, then ended the interview.
"This interview is over," he said, speaking off-camera.
Carson was also pressed on the same topic Sunday on ABC's "This Week" by host Martha Raddatz.
Carson told Raddatz that if a candidate wouldn't reject Islam's central tenets, "why in fact would you take that chance?"
He also said he'd listen to arguments that religion should be reason for probable cause to track the emails and phone calls of Syrian refugees and others from the Middle East.
"I personally don't feel that way, but I would certainly be willing to listen to somebody who had evidence to the contrary," Carson said. "I think that's one of the problems, we get to our little corners and we don't want to listen to anybody."