The frontrunner for the Republican presidential nomination tweeted there was “no chance” Mr Obama would defend him if he was similarly attacked.
South Carolina Senator Lindsey Graham has said Mr Trump should apologise.
The tycoon cancelled his appearance at a big Republican event on Friday.
His campaign team said he had pulled out of the Heritage Foundation because of a “significant business transaction” that needed his attention.
The criticism has been piling up since a man at Mr Trump’s rally in New Hampshire on Thursday night prefaced a question by saying Mr Obama was a Muslim and “not even an American”.
The supporter went on to say: “We have a problem in this country – it’s called Muslims.”
Mr Trump let it go unchallenged and within a few hours, Democratic frontrunner Hillary Clinton said his failure to denounce “hateful rhetoric” was “disturbing and wrong”.
And on Friday his Republican competitors for the nomination waded into the row.
“He’s playing into this hateful narrative and he has to set it right,” said Mr Graham, who said he would never question the president’s faith or patriotism.
Leaders have an “obligation” to correct such statements, said another Republican presidential hopeful, New Jersey Governor Chris Christie.
But Mr Trump hit back in a series of tweets: “Am I morally obligated to defend the president every time somebody says something bad or controversial about him? I don’t think so!
“If someone made a nasty or controversial statement about me to the president, do you really think he would come to my rescue? No chance!”
Mr Trump added that if he had challenged the man he would have been accused of interfering with his right to free speech.
President Obama, who has spoken openly about his Christian faith, was born to an American mother and Kenyan father in Hawaii.
But Mr Trump has been one of the leading sceptics, challenging Mr Obama in 2011 to produce his birth certificate to disprove rumours that he was born in Kenya, which the president did.
With more than a year until polling day, the businessman is ahead of his Republican rivals in the polls despite having no political experience
2016 hopefuls US candidate
- The early Republican frontrunner is Donald Trump
- Hillary Clinton will have learnt much from her failed campaign of 2008
- Florida senator Marco Rubio lost some right-wing fans by backing a bipartisan immigration reform package
- Wisconsin governor Scott Walker appeals to both the Republican establishment and the Tea Party
- Libertarian Rand Paul has his supporters – and enemies – among Republicans
- Veteran congressman Bernie Sanders is drawing huge crowds at his rallies
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