NEW YORK (AFP) - The head of the largest city police department in America on Wednesday (Aug 3) joined a growing list of people calling Donald Trump unfit to lead, saying the Republican mogul "scares the hell" out of him.

"Mr Trump scares me, scares the hell out of me to be quite frank with you," New York police commissioner Bill Bratton told CBS News a day after announcing that he would retire in September.

Bratton, who has spent 45 years in public service, said he did not "get" why the Republican nominee, whose home he has visited, commanded so much support.

"I'm amazed that veterans groups are so charmed by him," said the outgoing police chief, who himself served in Vietnam. "I've just watched this whole campaign and I just shake my head."

The internationally respected police officer chastised the tycoon's "lack of depth on issues", his lack of compassion, questioned his refusal to apologise to the parents of a Muslim American soldier killed in Iraq and said he believed he was not fit to be president.

"I wonder if he's ever actually taken a punch in his life," Bratton told CBS.

"It's a lot tougher being soft and understanding when you need to be soft. Like, the apology to this family," he said of bereaved parents of a fallen soldier who assailed Trump's portrayal of Muslim Americans.

"There's no compassion there, it's all about him and never about anybody else. Strange times," said the police chief.

Bratton said he had known Trump "for 20 odd years" and had been to his apartment, but was not close or intimate with him.

"I know many people who are, who are very strongly supportive of him, although they have questions. But again, it's troubling," he said.

President Barack Obama on Tuesday delivered a virtually unprecedented rebuke of a presidential nominee, calling Trump "unfit" to be president and called on Republican leaders to disown him.