Trump's lambasting of Darroch on Twitter - he called the ambassador "a pompous fool" and "a very stupid guy" and criticized outgoing British Prime Minister Theresa May - drew condemnation from both sides of the Atlantic.
May said Darroch's departure was "a matter of deep regret".
Farage said this week he was not the right man for the job, but on Wednesday, after Darroch's resignation, tweeted: "The right decision, time (to) put in a non-Remainer who wants a trade deal with America". In a resignationletter on Wednesday, Darroch said 'the current situation is making itimpossible for me to carry out my role as I would like.
The insults had been in response to the publication last Sunday of a collection of confidential e-mails Sir Kim had sent to government officials in London in which he described the Trump White House as "clumsy and inept".
TRT World spoke to Olly Barratt for more on the story.
"We don't really believe this administration is going to become substantially more normal", Sir Kim added.
Mrs May told parliament after the resignation was announced that she had spoken to Mr Darroch, who was due to leave his post anyway at the end of the year. "Johnson's failure to back the British Ambassador in Washington makes him vulnerable to accusations of being weak towards Trump", Ford said on Wednesday. On his state visit to the United Kingdom in June, Trump confirmed he told May how to leave the European Union, but admitted he did not know if the strategy would have worked.
In present circumstances a political appointee to Washington would be a mistake; it would send an unfortunate message to the Diplomatic Service and do nothing for morale.
Boris Johnson, expected to become the new prime minister, said nothing in Darroch's support.
"Good government depends on public servants being able to give full and frank advice", she said. "I do not know the Ambassador, but he is not liked or well thought of within the U.S. We will no longer deal with him".
The most popular theory - and it is just a theory - in the national press is that Darroch was targeted because he had supported Britain staying in the European Union and the leak would help push him out of the job before his term was to conclude at the end of this year.
"Boris Johnson - a former foreign secretary and he hopes to be the future prime minister - has basically thrown our top diplomat under the bus", he told the BBC.
There were also indications that Sir Kim chose to resign after he failed to win the backing of former foreign secretary Boris Johnson, who is leading the race to replace Ms.
There was praise too from Ireland with Secretary General of the Department of Foreign Affairs Niall Burgess tweeting best wishes to Kim Darroch on what he described as a sad day for all professional diplomats. "In a very hard period, he was showing his customary professionalism". "You're the very finest of us".
Like his predecessors, Darroch was a prominent figure in Washington, meeting frequently with high-level United States officials and hosting parties at the stately British Embassy. He has to stopping giving Trump what he wants without condition. "I'll never let another country's leader choose who represents the United Kingdom".