Announcing the criminal investigation on Friday, the Met's Assistant Commissioner Neil Basu said the alleged leak had damaged UK international relations, reports the BBC.
He urged whoever was responsible to turn themselves in and "face the consequences".
"I would say to the person or people who did this, the impact of what you have done is obvious.
"However, you are now also responsible for diverting busy detectives from undertaking their core mission," Basu said, adding that anyone with information about the alleged leak or those responsible should contact the police.
The investigation was launched by the Met Police Counter Terrorism Command, which takes national responsibility for investigating allegations of criminal breaches of the Official Secrets Act, according to Basu said.
The government had already opened an internal inquiry into the publication of the memos leaked to the Mail newspaper on July 7.
Following the leak, Trump said that Darroch "has not served the UK well", adding that "we're not big fans of that man and he has not served the UK well".
On Monday he escalated his response with a series of tweets criticising May and her handling of Brexit, the BBC reported.
"What a mess she (May) and her representatives have created... I do not know the ambassador, but he is not liked or well thought of within the US. We will no longer deal with him."
Darroch was also disinvited from a dinner held at the White House on Monday night for the Emir of Qatar.
The criticism led the Ambassador to step down on Wednesday, saying it was "impossible" for him to continue.
Dating from 2017 to the present day, Darroch in the leak emails 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."
They also said that rumours of "infighting and chaos" in the White House were mostly true.
( With inputs from IANS )