DONALD Trump was "wrong" to retweet videos produced by the "hateful" Britain First organisation, Prime Minister Theresa May has said.
But Mrs May indicated that she is not withdrawing her invitation for the US President to come to the UK on a state visit, despite widespread calls for the trip to be cancelled.
Mr Trump sparked outrage in the UK by retweeting three videos from the far-right group, purportedly showing violent acts by Muslims.
In response to a statement from Downing Street describing his actions as "wrong", the President issued a late-night tweet directed personally at the Prime Minister, saying: "@Theresa_May, don't focus on me, focus on the destructive Radical Islamic Terrorism that is taking place within the United Kingdom. We are doing just fine!"
In her first personal response to the furore, at a press conference during a visit to Jordan, Mrs May said that the UK and US worked closely together in the fight against terrorism.
And she added: "The fact that we work together does not mean that we are afraid to say when we think that the United States have got it wrong and to be very clear with them.
"I am very clear that retweeting from Britain First was the wrong thing to do."
Mrs May's rebuke to the president won applause from her Jordanian audience.
Asked whether she regarded Mr Trump as a fit person to be hosted by the Queen on a state visit, the Prime Minister said only: "An invitation for a state visit has been extended and has been accepted. We have yet to set a date."
She said: "Britain First is a hateful organisation. It seeks to spread division and mistrust among our communities. It stands in fundamental opposition to the values we share as a nation - values of respect, tolerance and common British decency.
"On the issue of radical Islamism, British Muslims are peaceful and law-abiding who have themselves been victims of acts of terror by the far right. There are those who conduct acts of terror in the name of Islam, but it is not in the name of Islam.
"As Prime Minister, I am very clear about the priority that I give to dealing with the challenge of the threat of terrorism ... and extremism from whatever source they come."
Asked whether she regarded the US president as a "supporter and enabler of far-right groups", Mrs May said: "We must all take seriously the threat that far-right groups pose both in terms of the terrorist threat that is posed by those groups and the necessity of dealing with extremist material which is far right as well.
"I've commented in the past on issues in the United States on this matter. In the United Kingdom we take the far right very seriously and that's why we ensure we deal with these threats and this extremism wherever it comes and whatever its source."
But she made clear that she wanted to maintain the UK's close relations with the US, regardless of Mr Trump's actions.
"This is a long-term special relationship that we have," said Mrs May.
"It is an enduring relationship that is there because it is in both our national interests for that relationship to be there.
"As Prime Minister, I am clear that that relationship with the United States should continue. I think it is in the interests both of the United Kingdom and the United States and of the wider world."
In the House of Commons, a string of MPs demanded the cancellation of Mr Trump's visit, with some saying the Government should demand that the president apologise to the British people.
Paul Flynn said Trump should be "charged with inciting racial hatred" if he came to the UK, while fellow Labour MP Naz Shah suggested the "commander-in-tweet" should be placed on the list of those banned from entering the country for promoting "the hate-filled ideology of fascism".
Shadow home secretary Diane Abbott said Mr Trump's online behaviour was "offensive to all decent British people".
Home Secretary Amber Rudd appeared to indicate her agreement to a call from Tory MP Peter Bone for the president to delete his Twitter account, telling MPs: "I'm sure many of us might share his view."
But Ms Rudd urged MPs to bear in mind the importance of the UK's special relationship with the US when voicing their concerns.
"The importance of the relationship between our countries and the unparalleled sharing of intelligence between our countries is vital," said the Home Secretary.
"It has undoubtedly saved British lives. That is the big picture here, and I would urge people to remember that."
The criticism of Mr Trump in Parliament came after Commons Speaker John Bercow granted a debate on an urgent question from Labour's Stephen Doughty, who said the President was "either a racist, incompetent or unthinking - or all three" to retweet the Britain First videos.
Mr Bercow made clear his own disapproval of the content of the videos shared by the president, telling MPs: "I thought the House would want urgently to express support for the victims of racism and bigotry and to denounce their purveyors."
In a flurry of social media posts on Wednesday morning, Mr Trump retweeted three posts by Britain First deputy leader Jayda Fransen to his 43.6 million followers, including footage from the Netherlands purporting to show a Muslim migrant attacking a man on crutches.
The retweets were made without comment, but were regarded by many in the UK as an effective endorsement of the far-right group.
Communities Secretary Sajid Javid said Mr Trump had "endorsed the views of a vile, hate-filled racist organisation that hates me and people like me".
The criticisms from London appear to have sparked the virtually-unprecedented social media rebuke by a head of state to the leader of a close ally, as Mr Trump used a late-night tweet to address Mrs May directly.
His comments prompted demands for the cancellation of the state visit, which was offered by Mrs May during her visit to Washington in January and is expected to take place next year.
London Mayor Sadiq Khan said Mrs May should withdraw the invitation and demand an apology on behalf of the British people, while Liberal Democrat leader Sir Vince Cable branded Mr Trump an "evil racist" who should not be given the honour of a state visit.
Shadow foreign secretary Emily Thornberry said Mr Trump's tweets were designed to "humiliate and belittle" the Prime Minister and had put the Queen in a "very difficult and invidious position" as his host for the planned state visit.
"If he comes next year, a year which is supposed to be a really happy year for the royal family, what on earth are people supposed to make of it?" asked Ms Thornberry.
She told BBC Radio 4's Today programme that Mrs May was "absolutely right" to criticise Mr Trump's initial tweet, adding: "There are Americans who are our friends, we share many values with America. But we don't share values with this man."
Fransen, who was convicted last November of religiously aggravated harassment for hurling abuse at a Muslim woman in a hijab, welcomed Mr Trump's rebuke to the PM, tweeting: "Well said Mr President! If Theresa May expressed as much outrage at the content of my videos as she has that Donald Trump retweeted them, we'd be a lot safer."
Right-wing US commentator Ann Coulter, thought to have been the source through which Mr Trump came across the Britain First posts, said the president was right to hit back at Mrs May.
"I think he has only given as good as he gets," Ms Coulter told the Today programme.
"I think he has been verbally attacked from the mother country for a lot longer than he has been attacking Britain."
Ms Coulter, who is one of only 45 people Mr Trump follows on Twitter, admitted she did not know who Britain First were and made no attempt to check the veracity of their videos, even by clicking on Fransen's social media biography, before retweeting them.
The Britain First posts included unverified videos titled "Muslim migrant beats up Dutch boy on crutches!" and "Muslim Destroys a Statue of Virgin Mary!"
The Dutch authorities said on Wednesday that the attacker in the first video was born and raised in the Netherlands.
