Despite the rift, Trump says he got along famously with Prince Harry during the banquet at the palace earlier this week.
"In fact, he spent a lot of time talking to Ivanka and talking to my family. I went up – he couldn't have been nicer. Couldn't have been nicer. I think he's great.'
"Did he say: 'Come on - do you think my wife's nasty?" Morgan asked Trump.
"We didn't talk about it … I was going to because it was so falsely put out there. And when you see that transcript and you see, it's the exact opposite of what they said," he replied.
Prince Harry seemed reluctant to be pictured with Trump in Buckingham Palace earlier this week. He did not follow other members of the Royal Family as they walked around the exhibition of US-themed artefacts with Donald Trump on Monday.
The incident follows comments from Trump to The Sun, when asked about Markle's previous comments about him.
"What can I say? I didn't know that she was nasty," he told the UK newspaper, later denying he'd said that.
Trump also told Morgan Prince Charles had given him an impassioned speech on the environment and climate change. "We were going to have a 15-minute chat, and it turned out into an hour and a half. He did most of the talking," Trump said.
Despite Prince Charles' best efforts, it appears Trump remains unconvinced: "I believe there is a change in weather and that it changes both ways," he told Morgan.
He called the state banquet at Buckingham Palace one of the highlights of his life and said his late mother "would have been proud" because she was a huge fan of the Queen of England.
He also revealed that, contrary to previous reports, the NHS will not form part of any trade deal between the US and the UK post-Brexit.
In the half-hour interview, he also said he is willing to go to war with Iran but is also willing to meet President Rouhani if that increases the chance of peace.
Morgan also questioned Trump about gun control in the US but the president hit back at the UK's "knife crime epidemic": "I read an article where everyone's being stabbed ... They said your hospital is a sea of blood - all over the floors."
