Trump, who in the past has expressed doubts about whether Obama was born in the United States, was told by a man at a town hall event on Thursday in Rochester, New Hampshire, that, "We have a problem in this country. It's called Muslims."
"We know our current president is one. You know he's not even an American," the man said.
Trump did not interrupt the man or challenge his contention in any way.
Obama is a Christian who as president has attended church occasionally.
White House Press Secretary Josh Earnest said it is no surprise that the incident happened.
"Is anybody really surprised this happened at a Donald Trump rally?" Earnest said.
He said "the people who hold these offensive views are part of Mr. Trump's base" and that Trump had shown "a willingness to countenance the offensive views of one member of his audience."
The billionaire developer's rise to the top of the Republican field in the race for a November 2016 presidential nominee has been impervious to the various controversies that erupt around him almost daily.
Whether this latest episode would prove to be an important turning point for Trump was yet to be seen.
At least one Republican rival, Senator Lindsey Graham of South Carolina, said Trump had crossed a line, creating a "defining moment" for his candidacy.
“If I were Donald Trump, I would go on national television and say 'I handled it poorly and if I had to do it all over again, I would challenge his question,'" Graham said on MSNBC. "It's OK to apologize."
Democratic candidate Bernie Sanders also demanded that Trump apologise to the President.
Trump in 2011 triggered a controversy by demanding that Obama show evidence that he was born in the United States. The Democratic president produced a longer form of his birth certificate that made clear he was born in Hawaii, not Kenya, as some of his critics have contended.
Democratic candidate Hillary Clinton criticised her Republican rival, saying that Trump's refusal to denounce the questioner's remarks "is disturbing, & just plain wrong".
Trump canceled a campaign event scheduled for Friday in South Carolina, saying he had an important business transaction to attend to.
Trump's reaction contrasted with how 2008 Republican presidential nominee John McCain dealt with a woman who called Obama an Arab at a McCain campaign event. McCain immediately stopped the woman and called Obama a decent family man with whom he has policy differences.
Democratic presidential candidates Hillary Clinton and Bernie Sanders both sharply criticized Trump on Friday.
"He knew or he should have known that what that man was asking was not only way out of bounds, it was untrue, and he should have, from the beginning, repudiated that kind of rhetoric, that level of hatefulness of a questioner in an audience that he was appearing before," Clinton told reporters in Durham, New Hampshire.
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