UK Prime Minister Theresa May has said Donald Trump was "wrong" to retweet posts from a British far-right group.
But she stressed the "special relationship" between Britain and the US was "in both our nations' interests" and should continue.
And she rejected calls to cancel a state visit by the US president.
Speaking on a visit to Jordan, she said: "An invitation for a state visit has been extended and has been accepted. We have yet to set a date."
Quizzed about Mr Trump's tweets, she said: "The fact that we work together does not mean that we're afraid to say when we think the United States has got it wrong, and be very clear with them.
"And I'm very clear that retweeting from Britain First was the wrong thing to do."
The BBC's deputy political editor John Pienaar said in his blog that "managing a calm and stable relationship with a president like Donald Trump looks like a task that's well beyond Theresa May - and arguably - anyone else".
Scotland's First Minister Nicola Sturgeon said she agreed with Theresa May's criticism of Mr Trump, but thought she should go further and halt plans for a state visit next year.
Mr Trump's retweeting of posts by a far-right group "risks legitimising those who want to spread fear and hatred," Ms Sturgeon said, and was "completely unacceptable".
Labour leader Jeremy Corbyn said he thought the state visit - which would see Mr Trump being hosted by the Queen at Buckingham Palace or Windsor Castle - was "unlikely to go ahead in the event that he does come to Britain".
He added that Mr Trump would be welcome to visit a mosque in Finsbury Park, in his north London constituency, to learn about the "diversity of our society and the sense of inclusivity".
And he offered the US president some advice on tweeting, saying it was best to "hold yourself back" and "restrict yourself to two or three tweets a day".
On Wednesday the US president retweeted three videos posted by the British far-right group.
When a Downing Street spokesman said he had been "wrong" to do so the president hit back:
.@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!— Donald J. Trump (@realDonaldTrump) November 30, 2017
An unnecessary controversy
By Anthony Zurcher, North America reporter
It's clear at this point that Donald Trump won't let a perceived slight or criticism go unanswered - even if it's from a supposed friend. Even if it's from the leader of the president's closest international ally.
So shock isn't exactly the right word to describe the reaction to Mr Trump's initially botched attempt to tell Theresa May to, in effect, mind her own business. This is just another example of the US president's self-described "modern-day presidential" use of social media, where Twitter is a cudgel for score-settling no matter the diplomatic cost.
When Mr Trump assumed the presidency, one of the first foreign dignitaries he received was Mrs May, and it appeared they formed a quick bond - briefly holding hands as they walked past the White House Rose Garden. Those bonds will now be tested in a spat over a few morning retweets of inflammatory videos.
It's a wholly unnecessary controversy, but the international consequences could be all too real.
The US and the UK are close allies and often described as having a "special relationship". Theresa May was the first foreign leader to visit the Trump White House.
The row was raised in the House of Commons on Thursday morning with Home Secretary Amber Rudd saying she hoped Mrs May's criticism "would have some impact on the president".
Responding to a call by MP Peter Bone for the president to delete his Twitter account, she said: "It's interesting to note [Mr Bone's] advice regarding Twitter accounts - I'm sure many of us might share his view."
Former Conservative minister Tim Loughton said if Twitter was "genuine in its commitment to fight hate crime online" it would delete the US president's account.
Veteran Labour MP Paul Flynn asked whether Mr Trump should be "charged with inciting racial hatred" if he comes to the UK, a view echoed by fellow Labour MP Naz Shah.
The UK's opposition party is Labour - their foreign affairs spokeswoman Emily Thornberry told BBC Radio 5 live that Donald Trump was "not America" and the government should "ignore" and "work round him".
The Archbishop of Canterbury Justin Welby, the senior bishop in the Church of England, said it was "deeply disturbing" that Mr Trump had "chosen to amplify the voice of far-right extremists".
The videos shared by Mr Trump, who has more than 40 million followers, were initially posted by Jayda Fransen, the deputy leader of Britain First, a group founded by former members of the far-right British National Party.
Ms Fransen, 31, has been charged in the UK with using "threatening, abusive or insulting words or behaviour" over speeches she made at a rally in Belfast.
At a White House briefing on Thursday, spokeswoman Sarah Sanders was asked if the president knew who Ms Fransen was when he retweeted.
She said: "No, I don't believe so. But I think he knew what the issues are and that is that we have a real threat of extreme violence and terrorism."
On Wednesday, 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", adding: "He is wrong and I refuse to let it go and say nothing."
London Mayor Sadiq Khan repeated his call the state visit to be cancelled, saying: "It beggars belief that the president of our closest ally doesn't see that his support of this extremist group actively undermines the values of tolerance and diversity that makes Britain so great."
After already condemning Mr Trump's actions on Wednesday, Brendan Cox - whose wife, MP Jo Cox, was murdered by a right-wing extremist who shouted "Britain first" before committing the act - told the US president to focus on problems in his own country.
You have a mass shooting every single day in your country, your murder rate is many times that of the UK, your healthcare system is a disgrace, you can’t pass anything through a congress that you control. I would focus on that. https://t.co/SNcqOZGvLQ— Brendan Cox (@MrBrendanCox) November 30, 2017
Britain's ambassador to the US, Kim Darroch, said he had raised concerns about the president's tweets with the White House.
British people overwhelmingly reject the prejudiced rhetoric of the far right, which seek to divide communities & erode decency, tolerance & respect. British Muslims are peaceful and law abiding citizens. And I raised these concerns with the White House yesterday.— Kim Darroch (@KimDarroch) November 30, 2017
America's ambassador to the UK, Woody Johnson, responded to comments from Twitter users who were mostly critical of Mr Trump's tweets.
I wanted you to #AskWoody. Thank you for your questions and feedback. I have relayed your concerns to Washington. The U.S. & UK have a long history of speaking frankly with each other, as all close friends do. Our relationship is strong, vital and enduring.— Ambassador Johnson (@USAmbUK) November 30, 2017
What did Trump retweet?
The first video purportedly shows a "Muslim migrant" attacking a young Dutch man on crutches. However, the claim in this tweet appears to have little substance.
A spokesperson from the Dutch Public Prosecution Service told the BBC that the person arrested for the attack "was born and raised in the Netherlands" and was not a migrant.
The Dutch embassy in Washington DC confirmed this on Twitter.
.@realDonaldTrump Facts do matter. The perpetrator of the violent act in this video was born and raised in the Netherlands. He received and completed his sentence under Dutch law.— Netherlands Embassy 🇺🇸 (@NLintheUSA) November 29, 2017
The second video retweeted by Mr Trump shows a man smashing a statue of the Virgin Mary.
This video was uploaded to YouTube in 2013. The man in the clip says: "No-one but Allah will be worshipped in the land of the Levant," which could place him in Syria.
The third video originates from the riots that took place in Egypt in 2013, and shows a man being pushed from the top of a building in Alexandria. In 2015, those involved in the incident were prosecuted, and one man was executed.
White House spokeswoman Sarah Sanders said on Wednesday that Mrs May and other world leaders knew that "these are real threats that we have to talk about".
"Whether it's a real video, the threat is real," she said.
What other reaction has there been?
Mr Trump's actions on Wednesday were criticised by both Democrats and Republicans.
Republican Senator John McCain said he was "surprised" at the president's tweets.
Meanwhile, Senator Orrin Hatch of Utah said that Mrs May was "one of the great world leaders", adding that he had "incredible love and respect for her".
“PM @theresa_may is one of the great world leaders, I have incredible love and respect for her and for the way she leads the United Kingdom, especially in the face of turbulence.” -OGH— Senator Hatch Office (@senorrinhatch) November 30, 2017
(Photo: Hatch at 10 Downing Street last Tuesday) pic.twitter.com/KQoH2LoRe9
Khizr Khan, the father of US soldier Humayun Khan who was killed in the Iraq war, told Today: "[Mr Trump] holds the hatred. He is an actor, he acts and fabricates these facts to exploit people, innocent people, that fall victim to his bigotry and he sees the benefit.
"We all need to unite ourselves, all decent people of the world, against the menace of terrorism."
So POTUS has endorsed the views of a vile, hate-filled racist organisation that hates me and people like me. He is wrong and I refuse to let it go and say nothing— Sajid Javid (@sajidjavid) November 29, 2017