Mr Trump said: “The ambassador has not served the UK well, I can tell you that.”
Emails from Sir Kim Darroch said Mr Trump’s White House was “uniquely dysfunctional” and “divided”.
Trade secretary Liam Fox told the BBC the leak was “unprofessional, unethical and unpatriotic”.
He said that whoever released the emails had “maliciously” undermined the defence and security relationship with the US, “the most important global relationship that we have”.
“I hope if we can identify the individual, either the full force of internal discipline – or if necessary the law – will be brought to bear because this sort of behaviour has no place in public life,” he said.
As the Foreign Office launched an investigation into the source of the leak to the Mail on Sunday, Mr Trump told reporters in New Jersey: “We’re not big fans of that man and he has not served the UK well.
“So I can understand it and I can say things about him but I won’t bother.”
Analysis: Damage is ‘considerable’
By James Robbins, BBC diplomatic correspondent
Just imagine if every heavily encrypted report to Whitehall from all UK ambassadors overseas was instantly available on your mobile.
The candour would cease immediately and they’d become ultra-bland and useless as a tool in policy-making.
So, damage in this case is considerable. There will be a large number of potential suspects.
Diplomatic telegrams are seen by scores, often hundreds of people – ministers and officials – across several departments. That is to ensure grown-up and private conversations can be had based on large amounts of source material.
Of course, there is damage to relations between the UK and the Trump White House too.
Mr Trump likes to dish out insults and criticism (remember his frequent belittling of Theresa May over Brexit, and his all out verbal attacks on the mayor of London) but he is pretty thin-skinned when the verbal arrows are aimed at him.
The one person who is not under suspicion in London is Sir Kim himself. After all, as his current political master, Mr Hunt, has made clear, he was just doing his job.
In the emails, the UK ambassador to Washington said: “We don’t really believe this administration is going to become substantially more normal; less dysfunctional; less unpredictable; less faction-riven; less diplomatically clumsy and inept.”
Sir Kim questioned whether this White House “will ever look competent” but also warned the US president should not be written off.
Dating from 2017 to the present day, the leaked emails said rumours of “infighting and chaos” in the White House were mostly true and policy on sensitive issues such as Iran was “incoherent, chaotic”.
Although the Mueller investigation later found allegations of conspiracy between the Trump campaign and Russia were not proven, Sir Kim’s emails said “the worst cannot be ruled out”.
Foreign Secretary Jeremy Hunt said the memos reflected a “personal view”, not that of the UK government.
Mr Hunt – who is seeking to become the next Conservative leader and prime minister – said it was the ambassador’s job to give “frank opinions” but they did not reflect the government’s view.
He said Mr Trump’s administration is “not just highly effective but the best friend of Britain on the international stage”.
Speaking on the same programme, Brexit Party leader Nigel Farage said if Boris Johnson becomes prime minister and the UK leaves the EU by 31 October, “people like” Sir Kim would “not be around”.
Asked about speculation that he might take on the diplomatic role, Mr Farage said: “I don’t think I’m the right man for the job”, adding that he was “not a diplomat”.
However, he said he “could be very useful” when dealing with the US administration.
Who is Sir Kim Darroch?
Sir Kim is the British ambassador to the US, which means he 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 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.