Trump, when asked about Cameron's criticism, said such comments indicate he is unlikely to have a good relationship with the British prime minister.
"The prime minister has made his views on Donald Trump's comments very clear. He disagrees with them," the spokesman said.
"He continues to believe that preventing Muslims from entering the US is divisive, stupid and wrong. He stands by his comments."
Asked who Cameron would prefer to be the next president, the spokesman said he would not comment on another country's election but said the British leader had been clear he would work with whoever won.
"He is committed to maintaining the special relationship," he said.
The spokesman said no meeting or call between Cameron and Republican candidate Trump was currently planned, but if one were proposed the prime minister would consider it.
"It looks like we are not going to have a very good relationship," Trump earlier told Britain's ITV television.
"Who knows, I hope to have a good relationship with him but he's not willing to address the problem either."
‘Not stupid’, says Trump
Trump insisted he was "not stupid... just the opposite", and denied he was divisive, calling himself "a unifier".
He said his campaign trail policies were just "suggestions", but said there was a "tremendous" problem with Islamic extremism.
The tycoon rejected claims he was anti-Muslim.
"Absolutely not. I am anti-terror," he said.
Trump said he had "many Muslim friends", one of whom told him: "'Donald you have done us such a favour: you have brought out a problem that nobody wants to talk about'."
Trump also blasted the new London Mayor Sadiq Khan, the first Muslim elected to run a Western capital city.
Khan, who took office on May 9, has branded Trump "ignorant" on Islam and claimed the tycoon was making the world more dangerous by alienating mainstream Muslims.
Trump said he was offended by Khan's denouncement.
"I think they were very rude statements and, frankly, tell him I will remember those statements. They are very nasty statements," Trump said.
"When he won I wished him well. Now, I don't care about him.
"Let's see how he does, let's see if he's a good mayor."
A spokesman for Khan on Monday branded Trump's views "ignorant, divisive and dangerous".
"It's the politics of fear at its worst and will be rejected at the ballot box," the spokesman said.
