Before his arrival, Trump declared the United Kingdom was in "turmoil", offering minimal support to the British PM over her handling of Brexit, expressing hope he would meet "friend" Boris Johnson during the visit.
Late previous year, May herself criticised Trump for retweeting a message by a member of a British far-right group, and the speaker of parliament has said the president would not be welcome to address the chamber.
"I think people are incensed at the way in which the European Union has behaved so arrogantly and basically shown that their long-term agenda is to be our masters rather than our servants". "She wanted to go a different route", Trump said, adding that May "didn't listen".
"The sense of betrayal in this country will be huge, and I very much hope that she's out of office before too long", Farage said.
At the Chequers news conference, Mr Trump said he had been responding to the Sun's question about Mr Johnson as a possible prime minister, adding: "He has been very nice to me".
In a wide-ranging interview he also blamed the mayor of London for high crime rates in the city and failing to tackle terrorism.
Khan and Trump have clashed in the past.More news: India adopts 'world's strongest' net neutrality norms
Mr Trump and Mr Khan have come to blows before, with the London mayor forced to call Mr Trump's claims about the capital untrue.
"But when they make you feel unwelcome, why would I stay there?" he told British newspaper, The Sun, on July 12. "He has done a bad job", he added. I think she's a terrific woman, and I think she's doing a terrific job.
May's critics have said, however, that she reins in criticism of Trump because of the strong desire of many in her Conservative Party to reach a trade deal with the United States after Brexit.
Demonstrator Claudia Boes said she was pleased at the turnout, adding "even outside the normal political circles this is an issue that affects people".
In an interview with Britain's Sun newspaper published late on Thursday, Trump said May's plans for a business-friendly Brexit would leave it too close to the European Union to allow a new trans-Atlantic trade deal to be struck.
"If signing up to the #Trump world view is the price of a deal, it's not worth paying". Instead, the White House only pointed to the positive things Trump said about May, with no mention of the tape.
Trump appeared unsure of what to do, forcing the Queen to walk around him to get onto his right hand side for the inspection.More news: World’s ugliest dog dies weeks after winning title
But Jacob Rees-Mogg told the Wall Street Journal the comments "cut through" the government's line on the Brexit deal.
However, the president also suggested that the dispute with Khan was personal and that the mayor had shown disrespect to him by allowing a 20-foot "Trump Baby" balloon - an orange likeness of Trump wearing a diaper - to fly over London protests during the president's visit.
"I said tremendous things, and fortunately we tend to record stories now so we have it for your enjoyment if you'd like it", Trump said.
"He thought she was great on North Atlantic Treaty Organisation today and is a really terrific person".More news: Trump Attacks CNN, NBC in Joint Press Conference With British Prime Minister