Boris Johnson should have stood up for the former UK ambassador during an ITV debate, Chancellor Philip Hammond has hinted.
Speaking on ITV's Peston programme, Mr Hammond said: "As a former foreign secretary, I would have felt the obligation to stand up for a first class civil servant who has done nothing wrong."
He added Sir Kim Darroch did a "good job" as the UK's man in Washington.
Mr Hammond reiterated the points put forward by the Government that the memos should not have been leaked, and that Sir Kim was expected to give an honest assessment of the country he was in.
"It is absolutely essential that these [diplomatic memos] are kept confidential," the 63-year-old said.
"But it's very important that governments are informed about what's going on in the capitals of their allies and partners.
"Many of us would be in a position, if things we have written in confidence, were to be published, that would put us in a very difficult position."
Mr Hammond was agreed with by the Tory MP, Sir Nicholas Soames.
Sir Nicholas told ITV News' Political Editor: "I have to say I didn't think that Boris behaved in the way a former foreign secretary, who understands the importance of relationships with ambassadors and the importance of ambassadors telling the truth about what they see and find.
"I thought he hung Kim Darroch out to dry last night."
Mr Hammond also said that he does not expect to serve in the Cabinet if Mr Johnson becomes prime minister, and signalled that he would become a "nightmare" to the front runner in the Tory leadership race over a no-deal Brexit.
Mr Hammond insisted he would use the Commons backbenches to "vigorously" battle any attempt at withdrawing from the EU without a deal.
Asked if he would jump before he was pushed if Mr Johnson takes the Tory crown in two weeks' time, Mr Hammond told Peston: "My expectation is that I will not be serving in the next administration.
"But, I want to say this because I read some stuff in the papers earlier this week about how I would be a nightmare on the backbenches.
"I will continue to argue vigorously against a no-deal Brexit.
"And I will certainly do everything I can to prevent a no-deal Brexit without parliamentary approval."
The pointed comments came after Mr Johnson was put on notice to expect a legal battle with former prime minister Sir John Major if he tries to suspend Parliament in order to force through a no-deal Brexit.
Former Conservative leader Sir John said it would be "utterly and totally unacceptable" for any British premier to shut down Parliament, and he would seek a judicial review if it happened.
Mr Johnson dismissed Sir John's "very odd" threat of being dragged through the courts, insisting that Parliament should accept its responsibility to deliver Brexit.
But he has refused to rule out proroguing Parliament to prevent MPs blocking a no-deal exit from the European Union on October 31.
The Tory leadership campaign frontrunner said: "What we are going to do is deliver Brexit on October 31, which is what I think the people of this country want us to get on and do.
"I think everybody is fed up with delay and I think the idea of now consecrating this decision to the judiciary is really very, very odd indeed.
"What we want is for Parliament to take their responsibilities, get it done as they promised that they would.
"They asked the British people whether they wanted to leave in 2016, the British people returned a very clear verdict, so let's get it done."
In order to prorogue Parliament, shutting it down until the next state opening, a prime minister would have to ask the Queen to formally allow it.
Although the Queen's decision could not be challenged, Sir John said the advice of the prime minister could be.
The monarch would be "in the midst of a constitutional controversy that no serious politician should put the Queen in the middle of", Sir John said.
"I for one would be prepared to go and seek judicial review to prevent Parliament being bypassed," he told BBC Radio 4's Today.