During the press conference, Mr Trump - who regularly accuses the USA media of "fake news" over its coverage of him - also clashed with a reporter from the American NBC network, describing it as "possibly worse than CNN".
In the interview, done earlier this week, Trump disparaged May's Brexit plan: "I would have done it much differently".
Mr Trump said the UK/US relationship was the "highest level of special" and both made much of our mutual trading levels. The president would not be drawn on how "brutal" Brexit suggestion had been.
The British pound has fallen and United Kingdom politicians are reeling from the tornado-blast of U.S. President Donald Trump, who has humiliated Prime Minister Theresa May, crushed hopes of a quick U.S. -U.K. trade deal and boosted May's opponents in the high-stakes battle over Brexit.
The i reports that 100,000 people took to the streets of London to protest against the visit of Mr Trump. "I actually told Theresa May how to do it but she didn't agree, she didn't listen to me", he said.
But he praised Mrs May as a "very tough negotiator" and described her as an "incredible woman" who is "doing a fantastic job".
"I guess when they put out blimps to make me feel unwelcome, no reason for me to go to London", he told The Sun, which is owned by his media ally, Rupert Murdoch, owner of Fox News in the United States.
Designing creative effigies of Trump was among the main measures adopted by the Londoners to express their anti-Trump protests.
"If they do that, then their trade deal with the USA will probably not be made".
As Trump and May held a press conference in the United Kingdom, the former was asked if he "changed his tune" after his interview with The Sun where he slammed May's policies and threatened to pull a U.S. -U.K trade deal off the table.
Speaking to the media in Co Mayo on Friday, Mr Varadkar said there are no plans for the U.S. president to visit Ireland.
President Donald Trump and Britain's Prime Minister Theresa May stand together upon Trump's arrival for a meeting at Chequers, the prime minister's country residence, near Ellesborough, northwest of London, July 13, 2018.
During a joint news conference with Ms May, Mr Trump said he had "a lot of respect for the Prime Minister". "I think he's got what it takes".
A large Scottish flag flew in front of the picturesque waterfront, as he told reporters there: "I think it's going to be great".
Trump praised May's Conservative Party rival and former Foreign Secretary Boris Johnson and said he would make a "great Prime Minister", adding "I think he's got what it takes".
Organizers said 250,000 people were protesting in London and other demonstrations were expected in cities around the country, including Windsor where the US President was due to have tea with Queen Elizabeth. He was heralded by military bands on his arrival at the 92-year-old monarch's home where her grandson Prince Harry married USA actress Meghan Markle in May.