Donald Trump had always been critical of the deal, claiming it had stopped short of curbing Iran's ballistic missile programme, and pulled the United States out of it on 8 May 2018.
It followed the announcement Sir Kim had chose to quit, saying his position had become "impossible" following the publication of cables in which he described the Trump administration as "inept" and "dysfunctional".
He said at a Tory leadership hustings in Bedfordshire: "In my view there is no threat to national security implied by the release of this material".
Nonetheless, he mentioned the Metropolitan Police had been in point of fact helpful that the newsletter of "these explicit paperwork... may maybe well also describe a prison offence and one which carries no public interest defence". That is what they are there for.
He said, "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". "That is my view".
His comments were roundly condemned by politicians and senior journalists, who accused the Met of the kind of "heavy-handed" approach more usually associated with totalitarian regimes.
He said: "These leaks damaged UK/US relations and cost a loyal ambassador his job so the person responsible must be held fully to account".
Mr Johnson said he had spoken to Sir Kim on Thursday to tell his disappointment over his resignation and the ambassador instructed him he had not watched the TV debate.
He mentioned Republican Senator Lindsey Graham, who said in a tweet on Wednesday that Darroch "always understood the strength of President Trump and referred to him as the "Terminator" who is indestructible and will most likely be reelected".
Last week the newspaper published other leaked memos from Darroch about Trump in which the diplomat described the US leader as "inept", "insecure" and "incompetent" and his administration as "uniquely dysfunctional".
Sir Kim stepped down as U.S. ambassador on Wednesday, asserting it modified into "very no longer going" for him to continue.
The Counter Terrorism Command, the unit responsible for investigating alleged breaches of the UK's Official Secrets Act, will lead the probe.
Assistant Commissioner Neil Basu said the Metropolitan Police had been given legal advice that there is "no public interest defence".
In his statement, Basu urged the person or people who leaked the memos to turn themselves in.
Per some Whitehall sources, Sir Kim determined to resign after Mr Johnson failed to fully toughen him all thru a TV debate on Tuesday night.