The foreign secretary tweeted: "@realDonaldTrump friends speak frankly so I will: these comments are disrespectful and wrong to our Prime Minister and my country."
It comes after a series of diplomatic messages sent by Britain's ambassador to the US, Sir Kim Darroch, were leaked at the weekend in which he described Mr Trump and his administration as "incompetent", "inept" and "insecure".
Mr Hunt confirmed that he would keep Sir Kim on should he win the leadership contest, adding: "Allies need to treat each other with respect as @theresa_may has always done with you.
"Ambassadors are appointed by the UK government and if I become PM our Ambassador stays."
He reiterated his support during a live televised leadership debate on ITV, but his opponent Boris Johnson refused to be drawn on the ambassador's future.
Mr Johnson said: "He [Donald Trump] was dragged into a British political debate.. I don't think that's the right thing to do... but let's face it, our relationship with the US is of fantastic importance."
He added that "he would not be so presumptuous" to guarantee the ambassador's future, saying he alone will make the decision on who to hire for politically sensitive jobs should he win the contest.
The leaked memos have provoked the ire of the US president, who tweeted that Sir Kim has "not served the UK well" and that Washington "will no longer deal with him".
Earlier on Tuesday, Mr Trump renewed his Twitter attack on Sir Kim - calling him "wacky", a "pompous fool" and a "very stupid guy".
He said the ambassador had been "foisted upon the United States" and is "not someone we are thrilled with".
Mr Trump wrote: "The wacky Ambassador that the U.K. foisted upon the United States is not someone we are thrilled with, a very stupid guy."
Prime Minister Theresa May also came in for more criticism, with Mr Trump having on Monday accused her of making a "mess" of Brexit.
The president tweeted: "He [Sir Kim] should speak to his country, and Prime Minister May, about their failed Brexit negotiation, and not be upset with my criticism of how badly it was handled.
"I told @theresa_may how to do that deal, but she went her own foolish way-was unable to get it done. A disaster!
"I don't know the ambassador but have been told he is a pompous fool.
"Tell him the USA now has the best Economy & Military anywhere in the World, by far and they are both only getting bigger, better and stronger.....Thank you, Mr. President!"
Mrs May told a cabinet meeting on Tuesday that it was "absolutely right" that they continued to give their full support to Sir Kim, and that the leaked messages did not reflect the closeness of the relationship with the US.
Shortly after Mr Hunt's tweets, US officials confirmed a meeting between International Trade Secretary Liam Fox and US Commerce Secretary Wilbur Ross had been cancelled.
However, a spokesperson from the US commerce department told Sky News: "The Department is working to reschedule today's meeting to a mutually agreeable time."
Tory leadership front runner Boris Johnson said he had a "good relationship" with Washington.
"I think it's very important that we have a strong relationship with our most important ally," he said.
"The United States is, has been, will be for the foreseeable future the number one political, military friend and partner."
The House of Commons' foreign affairs committee has launched an urgent inquiry into who leaked the ambassador's messages to the media.