Speaking Wednesday in London, he said American voters faced a decision "of hope over fear, unity over division."
"A choice of somebody who is trying to divide, not just your communities in America but who is trying to divide America from the rest of the world. And I think that, you know, that's not the America that I know and love."
He continued: "I'm hoping he's not the guy that wins."
London's first Muslim mayor
Khan is the first Muslim mayor of any Western capital.
Wednesday is his third day in office. His election Thursday made headlines around the world, and was followed almost immediately by a spat with the presumptive Republican presidential nominee, who has sparked outrage on the campaign trail with his comments on Islam.
Trump, who earlier proposed a temporary ban on Muslims entering the United States, suggested Monday he would make an "exception" for Khan.
The London mayor quickly shot back, rejecting the suggestion.
"I'm not exceptional," he told Amanpour. "So for Donald Trump to say, 'Oh, but Mayor Khan can be allowed, but not the rest,' is ridiculous."
He added: "The point is it's not just about me. It's about the message it sends from, you know, the greatest country in the world. I mean, what is the story of America? And I think, you know -- I think Donald Trump doesn't get the history of America."
Khan reiterated his belief that Trump's views of Islam are "ignorant."
"It is possible to be a Muslim and to live in the West. It is possible to be a Muslim and to love America," he said, adding that he himself loves the country.
"By giving the impression that Islam and the West are incompatible, you are playing into the hands of the extremists."
He continued: "Imagine that America has a sign saying no Muslims: What message does that send to Muslims around the world?"
Support for Clinton
Khan said he hoped that Trump's likely Democratic rival, Hillary Clinton, prevailed in the presidential race.
"I'm the father of two daughters. I'm a proud feminist in City Hall," he said.
"Just imagine the message it sends to my daughters and to girls around the world that the president of the United States is a woman. Not any old woman, a woman with the gravitas, with the experience, someone who is a unifier."
Clinton, who congratulated Khan for his win in a tweet Saturday, would be "an inspiration," he said.
"I'm quite clear in my mind who I want to be the president of the USA."
Khan told reporters earlier he would be happy for his politics to be a "template" for the American presidential race.
"London could have chosen the politics of fear ... chosen the option put forward by my opponents, and they chose not to. If that's a template for America, I'm really happy to help."
Speaking to Amanpour, he repeated his accusation that his Conservative rival, Zac Goldsmith, had "used the Donald Trump playbook" by targeting his religion during the bitterly contested London mayoral campaign.
"I've never hidden the fact that I'm a Muslim. And to try and seek to divide communities, to try and give the impression that our city may be less safe if I was the mayor ... I think that was rejected last week."
Comments on Islam spark outrage and petition
Trump's remarks about Islam on the campaign trail have provoked a heated response in the United States and abroad.
In the wake of the Paris terror attacks in November, he called for "a total and complete shutdown of Muslims entering the United States until our country's representatives can figure out what is going on."
He also proposed the creation of a database for Muslims and claimed that Britain has a "massive Muslim problem."
In the UK, a public petition to ban Trump from the country for hate speech attracted more than 500,000 signatures.
The petition prompted members of Parliament to hold a nonbinding debate on the issue in January, with one lawmaker opining, "I don't think Donald Trump should be allowed within 1,000 miles of our shore."