Foreign Secretary Jeremy Hunt, who is jousting with Boris Johnson to become the next prime minister, tweeted Saturday that the person responsible for the leak must be found and held responsible, but he differed with police over whether the publication of leaks is a possible crime.
Boris Johnson, frontrunner to be Britain's next prime minister, denied he was responsible for the resignation of the ambassador to Washington but admitted his comments had been a factor in the shock departure of one of the country's most senior diplomats.
Under the terms of the agreement - still supported by Britain, France and Germany - global sanctions on Iran were eased in return for Tehran accepting curbs on its nuclear programme.
The memos said rumors of "infighting and chaos" in the White House were mostly true.
"They can't articulate any "day-after" strategy; and contacts with State Department this morning suggest no sort of plan for reaching out to partners and allies, whether in Europe or the region", he wrote.
A British investigation into the leaking of confidential diplomatic memos is raising press freedom issues with a police warning that United Kingdom media might face a criminal inquiry if leaked documents are published.More news: Sudan military stop attempted coup, arrest involved security officers
Announcing the police inquiry, Counterterrorism police unit leader Neil Basu warned against any further publication of leaked documents.
In a cable sent afterwards, Darroch reportedly indicated there were divisions in Trump's team over the decision, and criticised the White House for a lack of long-term strategy.
Not all politicians, though, felt the police were wrong.
London's Metropolitan Police said on Friday its counter-terrorism command, which takes national responsibility for investigating allegations of criminal breaches of the Official Secrets Act, was leading the investigation.
Mr Hunt said he had been "disappointed" in Mr Johnson's response during the debate when he tried to pin him down on whether he supported Sir Kim following a furious tirade against the ambassador from Mr Trump.
"I would advise all owners, editors and publishers of social and mainstream media not to publish leaked government documents that may already be in their possession, or which may be offered to them, and to turn them over to the police or give them back to their rightful owner, Her Majesty's Government".More news: Ursula von der Leyen signals she will not reopen Brexit talks
"We know these documents and potentially others remain in circulation".
The cable was included in the second batch of leaked reports published by the Mail on Sunday newspaper, the first of which caused Darroch to resign earlier this week.
A spokesman for the newspaper said it was in the public interest and revealed "important information" on the UK's attempts to stop President Trump abandoning the Iran nuclear deal.
The chances the leak came after a foreign government hack have been ruled out. The perpetrator should face the consequences of their actions.
He added that it was "not news" that the United Kingdom and USA differ in their approach to preventing Iran acquiring a nuclear weapon.More news: Heidi Klum ties the knot with Tom Kaulitz in private ceremony
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