* Trump hits back after UK's May criticism
* President had retweeted posts from far-right UK party
* May says Trump was wrong as tweet causes UK outrage
* Calls for state visit offer to be withdrawn (Adds May comment, more reaction)
LONDON, Nov 30 (Reuters) - U.S. President Donald Trump sparked outrage in Britain on Thursday with a sharp rebuke of Prime Minister Theresa May on Twitter after she criticised him for retweeting British far-right anti-Islam videos.
As British politicians lined up to condemn Trump for sharing videos originally posted by a leader of a British far-right fringe group, Trump, in an unprecedented attack on one of America's closest allies, replied with an unrepentant message.
"Theresa ztheresamay, 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," he tweeted.
His truculent response caused anger in Britain, where there have been several major Islamist militant attacks this year, with one minister describing Trump's tweets as "alarming and despairing". London's Muslim mayor called for the withdrawal of an offer to him to make a state visit to Britain.
May, on a visit to Jordan, repeated her view, expressed earlier by her spokesman, that the U.S. leader was wrong for sharing anti-Muslim videos posted by Jayda Fransen, deputy leader of Britain First. But she did not directly respond to Trump's rebuke.
"I'm very clear that retweeting from Britain First was the wrong thing to do," May told reporters in Jordan. She said the group was a "hateful organisation" that sought to spread division and mistrust.
"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," May said. She added that Britain had a long-term, enduring relationship with the United States.
The British ambassador to the United States, Kim Darroch, said he had raised concerns with White House officials. "British people overwhelmingly reject the prejudiced rhetoric of the far right, which seek to divide communities & erode decency, tolerance & respect," he wrote on Twitter.
Fransen, who was convicted this month of abusing a Muslim woman and whose group wants to ban Islam, is facing further criminal charges of racially aggravated harrassment.
TRUMP AND MAY ONCE HAND-IN-HAND
Islamist militants have carried out several major attacks in Britain this year that have killed a total of 36 people, including a bombing in Manchester and two attacks on bridges in London in which victims were rammed with vehicles and stabbed.
Trump initially addressed his rebuke to a Twitter handle that was not May's, though he later retweeted to the British leader's correct account.
Always a pillar of Britain's foreign policy, the so-called "special relationship" with Washington has taken on added importance as Britain prepares to leave the European Union in 2019 and seeks new major trade deals.
Since Trump became president, May has gone out of her way to cultivate a good relationship with him.
She was the first foreign leader to visit him after his inauguration in January, and they were filmed emerging from the White House holding hands. She later said Trump took her hand in a gentlemanly gesture as they walked down a ramp.
But she angered his many critics in Britain then by extending an invitation to make a state visit to Britain with all the pomp and pageantry it brings including a formal banquet with Queen Elizabeth.
London's mayor, Sadiq Khan, said May should withdraw the offer of a state visit. "After this latest incident, it is increasingly clear that any official visit at all from President Trump to Britain would not be welcomed," Khan, who has clashed on Twitter with Trump, said.
British lawmakers held an urgent session to discuss Trump's tweets, with parliamentarians from across the political divide united in condemnation.
"By sharing it (the videos), he is either a racist, incompetent or unthinking, or all three," opposition Labour lawmaker Stephen Doughty said. Britain's Middle East minister Alistair Burt tweeted: "The White House tweets are both alarming and despairing tonight. This is so not where the world needs to go."
Despite repeated calls from opposition lawmakers to cancel the state visit, Home Secretary (interior minister) Amber Rudd said the invitation still stood although a timing had not been agreed.
Outside parliament, there was more harsh criticism from the likes of Brendan Cox, the husband of lawmaker Jo Cox who was murdered in 2016 by a far-right extremist and Justin Welby, the spiritual head of the Anglican Church.
The U.S. ambassador to London Woody Johnson wrote on Twitter he had relayed concerns to Washington. "The U.S. & UK have a long history of speaking frankly with each other, as all close friends do," he said.
The videos shared by Trump purported to show a group of people who were Muslims beating a teenage boy to death, battering a boy on crutches and destroying a Christian statue.
Reuters was unable to verify the videos. The Dutch embassy in Washington issued a Twitter comment on one of them, which Fransen had described as showing a "Muslim migrant" beating up a boy.
"zrealDonald Trump Facts do matter. The perpetrator of the violent act in this video was born and raised in the Netherlands," the embassy said. "He received and completed his sentence under Dutch law."
Britain First, a little-known party on the periphery of UK politics, welcomed Trump's retweeting of the videos to his 44 million followers, regarding it as an endorsement of their message.
"I'm delighted," said Fransen, whose own Twitter following increase by 50 percent in the wake of the furore to 78,000. She told Reuters Trump's retweets showed the president shared her aim of raising awareness of "issues such as Islam".
The White House defended the retweets by the Republican president, who during the 2016 U.S. election campaign called for "a total and complete shutdown of Muslims entering the United States", saying that he was raising security issues.
It repeatedly refused to be drawn into the content of the videos or whether Trump was aware of the source of the tweets.
"It's about ensuring that individuals who come into the United States don't pose a public safety or terrorism threat," White House spokesman Raj Shah told reporters aboard Air Force One. (Additional reporting by Estelle Shirbon, Elizabeth Piper and William James; Editing by Richard Balmforth)