"I would not advocate that we put a Muslim in charge of this nation. I absolutely would not agree with that," the retired neurosurgeon and Republican presidential candidate said Sunday on NBC's "Meet the Press."
Carson's comment comes on the heels of GOP front-runner Donald Trump taking a question from an campaign rally attendee who said, "We have a problem in this country; it's called Muslims."
Trump has since said he has many Muslim friends, but Sunday on CNN's "State of the Union" he added that the problem is "radical Muslims."
Carson, meanwhile, was asked Sunday whether a president's faith should matter to voters.
"I guess it depends on what that faith is," he said. "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."
Asked whether Islam is consistent with the Constitution, Carson said: "No, I don't -- I do not."
He said he is open, though, to voting for a Muslim for Congress.
"Congress is a different story, but it depends on who that Muslim is and what their policies are, just like it depends on what anybody else is," Carson said. "If there's somebody who is of any faith but they say things and their life has been consistent with things that will elevate this nation and make it possible for everybody to succeed and bring peace and harmony, then I'm with them."
Carson said he has no reason to believe President Barack Obama isn't a Christian. The man at Trump's rally had falsely claimed that Obama is a Muslim.
Sen. Amy Klobuchar, a Minnesota Democrat, was asked about Hillary Clinton's role in not squashing talk of President Barack Obama being a Muslim in their 2008 battle. But Klobuchar deflected the question and put it back on Carson.
"I think it's been pretty clear that Secretary Clinton has an excellent relationship with President Obama," Klobuchar told CNN's Chris Cuomo on "New Day" Monday. "I think the bigger issue here is you have Ben Carson basically saying a Muslim shouldn't be president, and I'd like him to say that to my staff member, who is Muslim, who set it up so I can be here today."
Carson's comments angered one of the two Muslims in Congress -- Minnesota Democratic Rep. Keith Ellison.
"For Ben Carson, Donald Trump, or any other Republican politician to suggest that someone of any faith is unfit for office is out of touch with who we are as a people," Ellison said in a statement. "It's unimaginable that the leading GOP presidential candidates are resorting to fear mongering to benefit their campaigns, and every American should be disturbed that these national figures are engaging in and tolerating blatant acts of religious bigotry."