British Prime Minister May, who is on a visit to Jordan, condemned British far-right group Britain First whose messages were re-tweeted by Trump as a "hateful organisation that seeks to spread division and mistrust" and represented everything that goes against "common British decency".
"The fact that we work together does not mean that we are afraid to say when we think the United States have got it wrong and be very clear with them. And, I am very clear re- tweeting from Britain First was the wrong thing to do," she said.
May, who was personally tagged on a follow-up tweet by Trump asking her not to focus on him but on the "destructive Radical Islamic Terrorism that is taking place within the United Kingdom", stressed that the UK remains at the forefront of "confronting terrorism, wherever it comes from".
The row dominated UK's Parliament proceedings with an urgent question on the issue in the House of Commons today.
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"President Trump was wrong to re-tweet videos from the far-right group Britain First," Rudd told MPs.
She, however, added that the intelligence-sharing between the two allies "undoubtedly saved British lives" and "that is the bigger picture here and I would urge people to remember that".
Several MPs demanded a proposed state visit next year on an invitation extended on behalf of Queen Elizabeth II be cancelled, with former Labour Cabinet minister Yvette Cooper saying Britain could not "simply roll out a red carpet and give a platform for the President of the United States" to "sow discord in our communities".
"As the mayor of this great diverse city, I have previously called on Theresa May to cancel her ill-judged offer of a state visit to President Trump. After this latest incident, it is increasingly clear that any official visit at all from President Trump to Britain would not be welcomed," he said.
The UK home minister seemed to indicate in the Commons that the visit remains effectively delayed indefinitely.
"An invitation for the visit has been extended and accepted, but the dates and the precise arrangements have yet to be agreed," Rudd said.
Another Tory MP Tim Loughton said if Twitter was "genuine in its commitment to fight hate crime online" it would delete the US President's account.
"I am sure that the chief executive of Twitter will have heard the interesting suggestions from (Mr Loughton) and we will leave it to them to decide what action to take," Rudd replied.
Other MPs branded Trump as "fascist", "either racist, incompetent or unthinking" and a spreader of "evil".
"He is wrong and I refuse to let it go and say nothing," Javid added.
The videos shared by Trump, who has more than 40 million Twitter followers, were initially posted by Jayda Fransen, the deputy leader of Britain First, a far-right group founded by former members of the far-right British National Party (BNP).
The 31-year-old, who faces two separate trials for religiously aggravated harassment and for using threatening and abusive language in the UK, posted on Twitter in capital letters: "Donald Trump himself has re-tweeted these videos and has around 44?million followers! God bless you Trump! God bless America!".
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 second video re-tweeted by Trump shows a man smashing a statue of the Virgin Mary, which was uploaded on YouTube in 2013.
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.