On Friday, Trump said on Twitter that he didn't suggest creating such a database but instead was answering a question from a reporter about the idea. However, he did not disavow the prospect of a database on social media or at an event Friday morning.
Civil liberties experts said a database for Muslims would be unconstitutional on several counts, while the libertarian Cato Institute's Ilya Shapiro said the idea also violates basic privacy and liberty rights.
Marci Hamilton, a Yeshiva University legal expert on religious liberty, said requiring Muslims to register appears to be a clear violation of the Constitution's protection of religious freedom.
"What the First Amendment does and what it should do is drive the government to use neutral criteria," Hamilton said. "You can use neutral criteria to identify terrorists. What it can't do is engage in one-religion bashing. That won't fly in any court."
Meanwhile, the Anti-Defamation League in New York called Trump's proposal "deeply troubling and reminiscent of darker days in American history when others were singled out for scapegoating."
The Council on American-Islamic Relations condemned as "Islamophobic" comments from both Trump and fellow GOP candidate Ben Carson, who on Thursday compared blocking potential terrorists posing as Syrian refugees from entering the U.S. to handling a rabid dog.
"If there's a rabid dog running around in your neighborhood, you're probably not going to assume something good about that dog," Carson said in Alabama. "It doesn't mean you hate all dogs, but you're putting your intellect into motion."
CAIR's Robert McCaw said in a statement, "Donald Trump and Ben Carson are contributing to an already toxic environment that may be difficult to correct once their political ambitions have been satisfied."
In New Hampshire on Friday, Carson said the U.S. should have a database on "every foreigner who comes into this country," but he rejected the idea of tracking U.S. citizens based on their religion.
"One of the hallmarks of America is that we treat everybody the same," he said. "If we're just going to pick out a particular group of people based on their religion, based on their race, based on some other thing, that's setting a pretty dangerous precedent."
The controversy followed attacks in Paris on Nov. 13 that killed 130 people and wounded hundreds more. The Islamic State group has claimed responsibility, elevating fears of attacks in the U.S. and prompting calls for new restrictions on refugees fleeing war-torn Syria.
Democratic front-runner Hillary Rodham Clinton took to Twitter Friday and challenged all Republican candidates to disavow Trump's comments. Vermont Sen. Bernie Sanders called Trump's words "outrageous and bigoted."
"This is shocking rhetoric," Clinton wrote. "It should be denounced by all seeking to lead this country."
Several did just that.
"You're talking about internment, you're talking about closing mosques, you're talking about registering people, and that's just wrong," Bush said Friday on CNBC.
A spokesman for Sen. Rand Paul, R-Ky., said the candidate "does not support databases based on one's religion."
Kasich, the Ohio governor, said requiring people to register with the federal government because of their religion "strikes against all that we have believed in our nation's history." Kasich had faced criticism following the Paris shootings for saying he would set up an agency with a mandate to promote what he called "Judeo-Christian values" overseas to counter Islamist propaganda.
Trump spoke Thursday a few hours after the House passed legislation essentially barring Syrian and Iraqi refugees from the United States. Senate Majority Leader Mitch McConnell, R-Ky., has slotted the bill for possible Senate consideration, though it's unclear whether the chamber could get enough votes to override a veto by President Barack Obama, who opposes the measure.
The unified pushback against Trump was rare. Republicans have vacillated in their handling of other inflammatory comments from him, wary of alienating his supporters but also increasingly concerned that he's managed to maintain his grip on the GOP race deep into the fall.
The first reference to a database for Muslims came in Trump's interview with Yahoo News published Thursday in which the billionaire real estate mogul did not reject the idea of requiring Muslims to register in a database or giving them special identification cards noting their religion.
"We're going to have to look at a lot of things very closely," Trump told Yahoo News.
According to Yahoo, he also suggested he would consider warrantless searches, saying, "We're going to have to do things that we never did before."
Asked by reporters Thursday night to explain his Yahoo comments, Trump suggested his response had been misconstrued. "I never responded to that question," he said.