In an interview with the UK's ITV, Mr Trump said he was not anti-Muslim, but "anti-terror".
He was reacting to remarks by UK Prime Minister David Cameron that his call for Muslims to be banned from the US was "divisive, stupid and wrong".
Mr Trump made the call last year, when he was not the party's front-runner.
He insisted that when he called for an immediate temporary ban on Muslims being allowed into America, there had been criticism only from politicians. Millions of people from all over the world had called in, he said, saying "Donald Trump is right".
Asked whether he would re-phrase those comments in the light of the controversy they caused, Mr Trump said: "It got people thinking. Whether it's good for me or bad for me, I don't really care."
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"Something very bad" was going on that people pretended didn't exist, Mr Trump said.
The world had a tremendous problem with radical Islamic terror, the New York billionaire said.
"If you look at it world-wide, the world is blowing up. And it's not people from Sweden that's doing the damage, okay?"
It is up to Muslims to turn in people they suspected of extremism, he told ITV.
"They have to work with the police. They're not turning them in. If they're not playing ball, it's not going to work out."
Referring to Mr Cameron's criticism, Mr Trump also said it looked like he was not going to have a good relationship with the UK prime minister.
He also criticised the new Mayor of London Sadiq Khan for calling him "ignorant".
The New Yorker is one of the least politically experienced nominees in US history, having never held elected office.
Many senior Republicans have refused to back him. All other Republican rivals have dropped out of the campaign.
Protests have plagued his campaign, with particular focus on his plan to build a wall on the Mexican border and deport 11 million undocumented immigrants.
On the Democratic side, Hillary Clinton appears closer to the nomination, having secured more delegates than Bernie Sanders.

Lee Devlin, 40, from Whitley Bay, died in hospital after being found with head injuries next to a walkway near the town's Metro station on 12 September.
The teenager, who cannot be named for legal reasons, pleaded not guilty to manslaughter at Newcastle Crown Court.
He was bailed to appear before Newcastle Crown Court for trial on 2 March.
A 16-year-old girl arrested on suspicion of causing grievous bodily harm remains on bail in connection with the death.
Two men, an 18-year-old arrested on suspicion of murder, and another, 19, on suspicion of wounding, have been released with no further action, Northumbria Police said.
The presumptive Republican candidate in the US presidential election, Donald Trump, has called on Muslims to work with the police and "turn people in".
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A 16-year-old boy has pleaded not guilty to the manslaughter of a North Tyneside man.