During her visit to Poland, Mrs May said that she expected leaks by former police officers in relation to Mr Green to be "properly investigated" and "taken seriously".
"I'm sure she didn't want him to go", Hunt said.
But Mr Hunt suggested Mr Green had been "pushed into a situation" where he said something he "didn't mean to say".
He also apologised for making writer Kate Maltby, who says he made inappropriate advances towards her in 2015, feel uncomfortable.
"This falls short of the Seven Principles of Public Life and is a breach of the Ministerial Code - a conclusion which has been endorsed by Sir Alex Allan, the Independent Adviser on Ministers' Interests".
Pictured (L-R) are Britain's First Secretary of State Damian Green, Britain's Prime Minister Theresa May and Britain's Parliamentary Secretary to the Treasury, and Chief Whip, Gavin Williamson, in central London, June 26, 2017.
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Damian Green leaves 10 Downing Street.
"I regret that I've been asked to resign from the government following breaches of the Ministerial Code, for which I apologise".
May accepted the resignation of First Secretary of State Damian Green, her right-hand man and an old friend, in a letter, with what she called "deep regret", adding that she was "extremely sad".
The cabinet secretary investigation said it was "not possible to reach a definitive conclusion" on whether Green's behavior with Maltby was appropriate wily, but added that her account was "plausible".
Ms Maltby is not commenting on Mr Green's resignation until she receives more details from the Cabinet Office.
Mr Hunt continued: "He lied on a particular incident, yes, I think lots of people who understand the context would appreciate why that might have happened but that doesn't make it any Cabinet ministers to the very highest standards of conduct, rightly, but I think we should remember that those are standards that would probably not apply in many other countries".
On the wider issue of dealing with instances of sexual harassment or bulling in Parliament, Mrs May said the leader of the House of Commons and leader of the House of Lords were working to ensure there was a proper grievance procedure.