"The current situation is making it impossible for me to carry out my role as I would like", he said in a statement.
Prime Minister Theresa May, who had given Darroch her full support, told Parliament after the resignation was announced that she had spoken to Darroch, who was due to leave his post anyway at the end of the year.
Darroch had been set to retire at the end of the year. "History shows that there are often bespoke procedures for filling the embassy in Washington, DC", he said.
Diplomatic memos he had written were released earlier this week questioning whether the White House "will ever look competent" under Mr Trump and referring to its "incoherent, chaotic" foreign policies.
But he did write that the Trump administration was unlikely "to become substantially more normal; less dysfunctional; less unpredictable; less faction riven; less diplomatically clumsy and inept". Trump also doubled down in attacking British Prime Minister Theresa May regarding Brexit, saying she "went her own foolish way-was unable to get it done". Trump had tweeted soon after he was elected in 2016, and that sentiment seemed to be part of the set-up that sent Darroch packing.
A formal civil service leak inquiry has been launched into how Sir Kim's messages were disclosed to the Mail on Sunday, but MPs called for the police to investigate. "We will pursue the culprit with all the means at our disposal", Foreign Office chief Simon McDonald told a committee of lawmakers, adding that police were involved in the investigation.
Chairman Tom Tugendhat said: "Even with states which could be described as hostile or not friendly, you know of no occasion where the head of state has refused to work with the British envoy?"
But he said the trans-Atlantic relationship was "so deep and so wide that it will withstand any individual squall". "They'll be more wary of how to do their job and more vulnerable", he says.
Darroch, a career diplomat who previously served as Britain's envoy to the European Union, had been accused by some Brexit supporters of being biased towards the bloc. "He should speak to his country, and Prime Minister May, about their failed Brexit negotiation, and not be upset with my criticism of how badly it was handled", Trump wrote on Twitter on July 9. This has brought home to me the depth of friendship and close ties between our two countries.
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