"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" on Sunday, Al Jazeera reported on Monday.
Carson, a devout Christian, said he thought a US president's faith should be "consistent with the constitution". He did not specify in what way Islam ran counter to constitutional principles.
America's largest Muslim civil rights group condemned Carson for his statement, which it said should disqualify him from the presidential contest because the US 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.
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 of support.
Sixteen Republicans are seeking the party's nomination for the US presidential election in November 2016.
The remarks by Carson followed a controversy that erupted when front-runner Donald Trump declined to challenge anti-Muslim comments made by a supporter on Friday.
The CNN/ORC poll showed Trump, a real estate mogul, continuing 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 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.
"I think Dr Carson needs to apologize," Graham said, saying 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.