When pressed, he also agreed with a proposal to bar people on terror watch lists from buying guns.
In an interview on ABC's "This Week," the Republican presidential front-runner said that in the wake of the Paris attacks by ISIS the United States must ramp up surveillance at mosques, particularly in New York City.
"I don't want to close up mosques, but things have to happen where -- you have got to use strong measures you're going to see buildings coming down all over New York City and elsewhere," he said.
Trump says he wants to keep a database of Syrian refugees -- who he said he'd prefer to keep out of the United States entirely.
"We have to be very, very vigilant," he said. "And a database would be fine for them, and (a) watch list is fine. We have to watch and see what they're doing."
He said he'd support a law barring people on terror watch lists who are "enemies of the state" from buying guns, despite his support for Second Amendment rights.
And Trump said he would bring back waterboarding, an interrogation tactic that he said "is peanuts compared to what they're doing to us."
And he said the rest of the presidential field -- particularly former Florida Gov. Jeb Bush, a Republican primary rival, and Democratic front-runner Hillary Clinton -- isn't strong enough to take on terrorists.
"Somebody like Jeb, and others that are running against me -- and by the way Hillary is another one; I mean, Hillary is a person who doesn't have the strength or the stamina, in my opinion, to be president," Trump said. "We're dealing with very, very strong people. And you need strength and you need stamina."
Trump also accused residents of New Jersey of cheering the September 11, 2001, fall of the World Trade Center towers.
"There were people that were cheering on the other side of New Jersey, where you have large Arab populations," Trump said. "They were cheering as the World Trade Center came down. I know it might be not politically correct for you to talk about it, but there were people cheering as that building came down -- as those buildings came down. And that tells you something."
Trump also declined to rule out a potential third-party bid for the White House, asked about it even as he continues to dominate GOP polls.
"I'm going to have to see what happens. I will see what happens. I have to be treated fairly," he said. "When I did this, I said I have to be treated fairly. If I'm treated fairly, I'm fine. All I want to do is (have) a level playing field."