Appearing on ITV’s Good Morning Britain, he aimed a broadside at Mr Johnson for avoiding a live TV debate.
The foreign secretary said: “It is very disrespectful to say you are not going to appear on any head-to-head debates in the two weeks before they have actually voted.
“If you are saying the only debates I will do is after people have voted, it’s not showing respect. People need to know what we are going to do.”
But Mr Hunt refused to attack Mr Johnson over his personal life, after his domestic row with partner Carrie Symonds dominated headlines over the weekend.
“I think that what happens in people’s personal lives is completely irrelevant,” he said, “given the constitutional crisis [over Brexit].
“People want to know: what are our answers on Brexit?”
Pledging to leave by the EU’s Halloween deadline with or without a deal, Mr Hunt said: “I just want to get on with this. Look at my record. I voted three times to leave the EU [through Theresa May’s Withdrawal Agreement].
“I argued the whole time we needed to get a better deal on the backstop and we would have left the EU by now [if we had].
“There’s no magic potion. This is going to be very difficult. But it’s doable because the EU want to try and resolve this with a deal.
“If we get to October 31 and Parliament has not taken no-deal off the table, then with a heavy heart I would leave.
“Who is the Prime Minister they can trust to go to Brussels and get a deal which can get through Parliament?”
Addressing Mr Johnson, he added: “I promise Boris Johnson the fight of his life. He’s going to have that and he’s going to lose.”
Mr Hunt was also quizzed about his views on abortion. He has argued in favour of halving the time limit from 24 weeks to 12.
He said: “As Prime Minister I will not seek to change the law on that issue. As far as that is concerned, that is it.”
Asked why he holds those views, Mr Hunt refused to answer, and repeated: “It’s nothing to do with what I want to do as Prime Minister. I don’t seek to change that law.”
It came as Mr Johnson also repeated his determination to deliver Brexit by Halloween, in what will be seen as an attempt to refocus attention away from his private life.
Mr Johnson vowed "we are not going to bottle it". Mr Hunt had previously attacked Mr Johnson over reports he was "bottling" the live TV debate before postal ballots are returned.
He wrote in his Daily Telegraph column: "We must leave the EU on Oct 31 come what may. It will honour the referendum result, it will focus the minds of EU negotiators."
He added: "It is absolutely vital that we keep our eyes on the prize. It has been a long and parching march - but the oasis is finally in sight.
"We are just over four months away from the date on which, by law, we must leave the EU; and this time we are not going to bottle it. We are not going to fail.
"This time we are not going to shrink in fear from the exit, as we have on the last two occasions."
He added that he was "bursting with impatience" to "shout our (the Conservatives') message from the rooftops" as party leader.
"I have done it before, victoriously, and am, of course, bursting with impatience to do it now, as leader of our party. We can win this argument big time - but first we must leave on October 31.”
Meanwhile, defence minister Tobias Ellwood has said "a dozen or so" Conservative MPs could support a vote of no confidence in the government to stop a no-deal Brexit.
He told BBC1's Panorama programme: "I believe that absolutely is the case. I think a dozen or so Members of Parliament would be on our side, would be voting against supporting a no-deal and that would include ministers as well as backbenchers.”
Mr Johnson's campaign for Number 10 has been rocked by the revelations that officers were called to the home he shares with partner Ms Symonds by a neighbour who claimed to have been "frightened and concerned" after hearing shouting, "a loud scream" and banging coming from the property.
A poll carried out after the reports emerged suggested Mr Johnson's support among Tory voters had dropped by more than half, while among the general electorate it indicated he had slipped into second place behind Mr Hunt.
When asked at the hustings about the incident, Mr Johnson said people "are entitled to ask about me and my determination, my character and what I want to do for the country".
But he would not elaborate on the police visit, preferring to talk about his time as mayor of London instead.
Tory grandee Sir Malcolm Rifkind rejected a suggestion that Mr Johnson should be allowed to draw a distinction between his private and public life and said his refusal to explain the situation with even a short comment was a "lack of judgment".
On Saturday night, the neighbour who rang the Metropolitan Police went public after suggestions that his recording of the row had been leaked to the Guardian with political intent.
Tom Penn said the allegations were "bizarre and fictitious", explaining in a statement to the paper that he dialled 999 after hearing shouting coming from his neighbour's flat.
Additional reporting by Press Association.