In a campaign statement, he said a “total and complete” shutdown should remain until the US authorities “can figure out” Muslim attitudes to the US.
At a rally in South Carolina hours later, frontrunner Mr Trump repeated the pledge, to loud cheers.
Criticism from the White House and other Republicans was swift.
Mr Trump’s comments were contrary to US values and its national security interests, a statement from the White House said.
Republican Jeb Bush, also running for president, said the New York businessman was “unhinged”.
Mr Trump’s statement was delivered as the US comes to terms with its deadliest terror attack since 9/11.
Last week a Muslim couple, believed to have been radicalised, opened fire and killed 14 people at a health centre in San Bernardino.
California attackers had target practice
On Sunday, President Barack Obama made a rare Oval Office address in response to the attack and warned against the US falling prey to divisiveness.
Mr Trump’s statement to reporters on Monday said polling by the Center for Security Policy, a conservative think-tank, indicated that 25% of Muslims in the US believed violence against America was justified.
“Without looking at the various polling data, it is obvious to anybody the hatred is beyond comprehension. Where this hatred comes from and why, we will have to determine.
“Until we are able to determine and understand this problem and the dangerous threat it poses, our country cannot be the victims of horrendous attacks by people that believe only in Jihad, and have no sense of reason or respect for human life.”
Again and again during his campaign, Donald Trump has grabbed the Republican Party by the neck and dragged it to the anti-immigration, nativist right.
With his call for a halt to all Muslim entry into the US – reportedly even for simple tourism – he has set down yet another marker that will force his fellow candidates to stand with him or risk his dismissive ire.
Mr Trump was ridiculed when he launched his White House bid by accusing Mexico of sending its “rapists and criminals” into the US, and yet the New Yorker shot up in the polls.
He was denounced after suggesting that some mosques should be closed and Muslims in the US monitored, but he solidified his status as the party’s frontrunner. In the end his hardline positions became largely accepted, if not embraced, by his fellow candidates.
Now he is being roundly condemned for his border-closing proposal. Will this finally be a bridge too far? Perhaps. Early reports, however, show his supporters welcoming this latest pronouncement. Such is the state of the 2016 presidential campaign season so far.
When asked by The Hill if that included Muslim Americans who may currently be abroad, his spokeswoman said: “Mr Trump says everyone.”
The director of the Council on American Islamic Relations, Nihad Awad, said Mr Trump sounded like the leader of a lynch mob rather than a great nation.
Soon after his statement was released, Mr Trump’s Republican rival Ben Carson called on all visitors to the US to “register and be monitored” during their stay.
But his spokesman added: “We do not and would not advocate being selective on one’s religion.”
Another Republican presidential hopeful, Senator Lindsey Graham, urged all those running to condemn Mr Trump’s remarks, which they did.