Trump took aim at Darroch in a series of Twitter tirades over the last few days, calling him a "stupid guy" and a "pompous fool" after UK media published a confidential memo in which the British diplomat described the president's administration as "uniquely dysfunctional", the Efe news reported.
In a letter, Darroch wrote: "Since the leak of official documents from this embassy there has been a great deal of speculation surrounding my position and the duration of my remaining term as ambassador.
"I want to put an end to that speculation.
"The current situation is making it impossible for me to carry out my role as I would like."
He added: "Although my posting is not due to end until the end of this year, I believe in the current circumstances the responsible course is to allow the appointment of a new ambassador.
"I am grateful to all those in the UK and the US, who have offered their support during these difficult few days. "
Sir Simon McDonald, permanent under-secretary at the Foreign and Commonwealth Office, said he accepted Darroch's resignation with "deep personal regret".
He said the outgoing ambassador had behaved with dignity and professionalism over the last few days.
Speaking to MPs, the UK's outgoing Prime Minister Theresa May, also expressed her regret.
"Sir Kim has given a lifetime of service to the United Kingdom and we owe him an enormous debt of gratitude," she said.
Darroch's departure came not long after Boris Johnson, the former Foreign Secretary who looks most likely to take over from May in a Conservative Party leadership battle, fell short of offering the British ambassador his support during a televised debate with his fellow contender, current foreign secretary Jeremy Hunt.
Sir Alan Duncan, a Conservative Foreign Office minister, heaped scorn on Johnson's actions, describing his refusal to back Darroch as "contemptible and not in the interests of the country he hopes to lead."
In a volley of angry tweets over the last two days, Trump levelled personal criticism at the ambassador.
"The wacky Ambassador that the UK foisted upon the US is not someone we are thrilled with, a very stupid guy," he said Tuesday.
"I don't know the Ambassador but have been told he is a pompous fool."
On Monday, Trump said that he would no longer be dealing with the British ambassador.
May was not spared a verbal attack either, as the US President once again criticised her handling of Brexit, saying she had ignored his advice and instead went her "own foolish way".
The row, the latest to jeopardize the so-called special relationship, emerged after the Mail on Sunday published the leaked emails.
Hunt also expressed his regret over Darroch's resignation.
He tweeted: "Deeply saddened to hear of the resignation of Sir Kim Darroch.
"Standing up for Britain means standing up for the finest diplomats on the world. It should never have come this."
The debacle escalated just as the UK's international trade secretary, Liam Fox, was visiting Washington.
Darroch decided not to accompany the cabinet minister to a meeting with Trump's daughter and advisor Ivanka Trump, a duty the UK ambassador would normally fulfill.
( With inputs from IANS )