Sir Nigel resigned after confidential diplomatic cables were leaked in which he criticised the Trump administration.
Speaking to Andrew Pierce on LBC, Mr Grieve said: "I think we're heading for a very bad place with Boris Johnson and that for me was quite a defining moment of an individual who has no responsibility except to himself".
The ambassador said his position had become "impossible" after the US President denounced him as a "very stupid guy" and a "pompous fool" on Twitter.
He lamented that the government was "effectively justifying state action against the media", adding that police should go after the leaker, if they breached the Official Secrets Act, but not the media. You can stop this now.
The statement also said there has been damage caused to Britain's worldwide relations, and there would be clear public interest in bringing the person or people responsible to justice.
British lawmakers are warning that the country is at risk of turning into a "police state", as police tell media outlets not to publish leaked government documents after a row that led to the resignation of the UK's ambassador to the US.
The announcement came as Tory leadership front runner Boris Johnson admitted his reported lack of support for Sir Kim had been a factor in the envoy deciding to step down.
He urged the culprits: "turn yourself in at the earliest opportunity, explain yourself and face the consequences".
"The publication of leaked communications, knowing the damage they have caused or are likely to cause may also be a criminal matter", the statement from Basu said.
In the Commons on Thursday, the Foreign Office minister Sir Alan Duncan said an internal Whitehall inquiry had found no evidence the leak was the result of computer hacking.
Sir Kim's resignation prompted widespread support for him - as well as criticism of Tory leadership frontrunner Mr Johnson.
"Some people just told me. he actually said very good things about me", Trump said.
"(Darroch) said that what somebody had relayed to him had certainly been a factor in his resignation", he told BBC television in a testy interview.
Trump earlier had publicly blasted the former British diplomat after reports unveiled details of Darroch's telegrams assessing the USA government and Trump's administration.