"I would not advocate that we put a Muslim in charge of this nation. I absolutely would not agree with that," Carson told NBC's "Meet the Press."
The remarks by Carson, who is near the top of opinion polls for the crowded field of Republican candidates, followed a controversy that erupted when front-runner Donald Trump declined to challenge anti-Muslim comments made by a supporter on Friday.
Carson, a Christian who says he got the idea for his tax proposals from the Bible, said he thought a U.S. president's faith should be "consistent with the Constitution."
Asked if he thought Islam met this bar, the retired neurosurgeon said: "No, I do not."
America's largest Muslim civil rights group condemned Carson for his statement, which it said should disqualify him from the presidential contest because the U.S. Constitution forbids religious tests for holding public office.
"It's beyond the pale and he should withdraw," said Council on American-Islamic Relations spokesman Ibrahim Hooper.
Minnesota Democratic Representative Keith Ellison, the first practicing Muslim elected to Congress, said: "It’s unimaginable that the leading GOP presidential candidates are resorting to fear mongering to benefit their campaigns."Ellison added in his statement that "every American should be disturbed that these national figures are engaging in and tolerating blatant acts of religious bigotry."
Trump: 'I love Muslims'
Meanwhile Donald Trump has stated he "loves Muslims" after sparking criticism for not correcting a man who verbally attacked Muslims at a speaking event and asked Trump, "When can we get rid of them?"
Trump was asked about the exchange at an event in Iowa by a school student.
"I consider Muslim Americans to be an important asset to our country and society. Would you consider putting one in your cabinet, or even on your ticket?" a student asked Trump.
"Oh, absolutely. No problem with that," Trump, telling a reporter outside the event: "I love the Muslims. I think they're great people."
Carson third in poll
Carson had been rising in polls, although he gave up some ground in a CNN/ORC poll released on Sunday, slipping to third place from second with 14 percent support. Sixteen Republicans are seeking the party's nomination for the presidential election in November 2016.
The CNN/ORC poll showed Trump, a real estate mogul, continued to lead the contest with the support of 24 percent of registered voters, down from 32 percent in a previous poll.
Former Hewlett-Packard Chief Executive Officer Carly Fiorina surged into second place with 15 percent support, just above Carson.
Carson's comments drew scorn from Republican Senator Lindsey Graham, another presidential candidate, although one who garnered under 1 percent support in the CNN/ORC poll.
"I think Dr. Carson needs to apologiSe," Graham said, adding the comments were particularly offensive to US soldiers who are Muslim.
But other Republicans tread more softly around the issue, highlighting its salience among some voters. Senator Rand Paul of Kentucky, who had 4 percent support in the CNN/ORC poll, told CBS' "Face the Nation" that a president's religion should not matter but he understood the rise of anti-Islamic sentiment because "we were attacked by people who were all Muslim."
A Des Moines Register/Bloomberg poll conducted in January in Iowa, the first state to vote in the nominating contest, showed 39 percent of Republicans saw Islam as inherently violent. Thirteen percent of Democrats held that view.
Trump, asked on NBC's "Meet the Press" on Sunday if he would accept a Muslim president, replied: "Some people have said it already happened."
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