In withering comments on Twitter, Mr Trump also lashed out at Theresa May, saying it was "good news" that Britain would soon have a new prime minister.
Sir Kim, in emails leaked on Sunday, described Mr Trump's administration as dysfunctional and inept.
Mrs May said she had "full faith" in Sir Kim but did not agree with him.
Her spokesman had described the leak as "absolutely unacceptable", and said the prime minister's office had made contact with the White House.
Trump diplomat row: Police urged to open investigation
Mr Trump had already responded to the leaked emails by saying "we're not big fans of that man and he has not served the UK well".
In Monday's series of tweets he said Sir Kim was "not liked or well thought of within the US".
Apparently angered by Mrs May's support for the ambassador, he again lashed out at her handling of Brexit, describing it as "a mess".
The US state department declined to comment on President Trump's remarks.
Is ambassador now 'persona non grata'?
Analysis by James Landale, BBC diplomatic correspondent
By saying he won't deal with Sir Kim Darroch any more, Donald Trump is apparently all but declaring the ambassador to be persona non grata. That is the formal legal process by which a host government expels a foreign diplomat.
The key question now is what the president means by the word "deal". If the royal "we" used by Mr Trump means that his entire administration will no longer deal with Sir Kim or any of his staff then the British government may have to decide to fast track the retirement of their man in Washington.
Sir Kim, who is an honourable man and was stepping down anyway in a few months, may decide to resign. If, however, Mr Trump merely means he won't deal personally with Sir Kim then the ambassador may stay on until the new prime minister can make his own appointment.
This all presents the British government with an awkward dilemma - to buckle under US pressure and bring Sir Kim home, risking accusations of abject weakness, or to stand firm and defend their ambassador for doing his job and telling the truth as he sees it, risking even further damage to the UK/US relationship.
What did the leaked emails say?
He questioned whether this White House "will ever look competent" but also warned that the US president should not be written off.
The emails, dating from 2017, said rumours of "infighting and chaos" in the White House were mostly true and policy on sensitive issues such as Iran was "incoherent, chaotic".
Sir Kim said Mr Trump was "dazzled" by his state visit to the UK in June, but warned that his administration would remain self-interested, adding: "This is still the land of America First."
Who is Sir Kim Darroch?
Sir Kim represents the Queen and UK government interests in the US.
Born in South Stanley, County Durham, in 1954, he attended Durham University where he read zoology.
During a 42-year diplomatic career, he has specialised in national security issues and European Union policy.
In 2007, Sir Kim served in Brussels as the UK permanent representative to the EU.
He was the prime minister's national security adviser between 2012 and 2015, dealing with issues such as the rise of the Islamic State group in Iraq and Syria, Russian annexation of Crimea, the nuclear threat from Iran and the collapse of government authority in Libya.
He became ambassador to the US in January 2016, several months before Donald Trump became president.