Boris Johnson and Jeremy Hunt have clashed on Brexit and UK relations with Donald Trump in a lively and occasionally bad-tempered TV debate.
Mr Hunt accused his rival of not being willing to "put his neck on the line" by saying he would quit as PM if he did not hit the 31 October deadline.
Mr Johnson said he admired his rival's ability "to change his mind" so often - a dig at the fact Mr Hunt voted Remain.
Mr Johnson declined to condemn Mr Trump for his response to the emails row.
He refused to confirm whether he would keep the UK's top diplomat in the US, Sir Kim Darroch, in his post until his scheduled retirement in December, after Mr Trump said he was no longer prepared to deal with him.
The US president has lambasted Sir Kim, and criticised Theresa May, after the diplomat described the White House as "inept and dysfunctional" in leaked cables.
While stressing the value of the "special relationship" with the US, Mr Johnson insisted that only he, as prime minister, would take "important and politically sensitive" decisions such as who should represent the UK in the US.
During the first head-to-head debate of the leadership campaign, the two clashed over their different Brexit strategies, political styles and why they were best equipped to be prime minister.
The exchanges were pointed and personal in nature at times, with former Mayor of London Mr Johnson dismissing his opponent's "managerial" style of politics and accusing him of flip-flopping on certain issues.
Foreign Secretary Mr Hunt said the UK needed a leader not a "newspaper columnist" - a reference to his rival's work for the Daily Telegraph.
He joked that he admired Mr Johnson's "ability to answer the question", adding: "He puts a smile on your face and you forget what the question was, a great quality for a politician but not necessarily a prime minister."
After an opening speech from each contender, the foreign secretary immediately went on the attack over Brexit, pressing his rival on whether he would quit Downing Street if he failed to take the UK out of the EU by 31 October.
He said by failing to answer the question, Mr Johnson - who previously said the deadline was a "do or die" issue for him - showed he was motivated by personal ambition not leadership.
"It is not do or die," Mr Hunt said. "It is Boris in Number 10 that matters."
Accusing his rival of not being straight with the electorate, he said: "Being prime minister is about telling people what they need to hear, not just what they want to hear."
Mr Johnson, in turn, said it was clear his rival was "not absolutely committed" to the deadline himself, branding him "defeatist".
He urged Mr Hunt to guarantee that Brexit would happen by Christmas, adding that the EU would not take a "papier mache deadline" seriously.
"If we are going to have a 31 October deadline, we must stick to it," he said. "The EU will understand we are ready and will give us the deal we need.
"I don't want to hold out to the EU the prospect that they might encourage my resignation by refusing to agree a deal.
"I think it is extraordinary we should be telling the British electorate we are willing to kick the can down the road.
"I would like to know how many more days my opponent would be willing to delay."
Both men have said they would be prepared to leave the EU without a deal, but Mr Johnson has been far more relaxed about the impact that could have.
Mr Hunt suggested his rival was "minimising the risk of a no-deal Brexit" and "peddling optimism", but Mr Johnson said the UK had had a "bellyful of defeatism" and the UK could look forward to a bright future outside the EU.
The pair also disagreed over whether they might be prepared to suspend Parliament to force through a no-deal exit - so-called prorogation.
While Mr Hunt categorically ruled this out, Mr Johnson said he would "not take anything off the table".
Both teams will leave Salford content with their candidates' performance.
The gaffe prone former foreign secretary avoided slipping on any banana skins, and managing not to commit on some of the more controversial issues before him.
And the current foreign secretary managed to land his blows on his opponent.
There was perhaps though no jaw dropper, no moment that turned this race upside down.
Mr Johnson arrived the favourite and leaves in the same position. Mr Hunt turned up keen to show that he is ready to use sharp elbows to scrap and to make himself heard with attacks on his rival that are a contrast to his normal careful style.
Their respective status as the front runner and challenger may not have changed.
Yet while Jeremy Hunt may not, from this performance alone, manage to stop Boris Johnson's journey to No 10, he has at least shown that if he gets there, he is likely to face a very tricky time.
On the escalating diplomatic row with the US, Mr Hunt said the president's criticism of Sir Kim Darroch had been ill-judged and he would, if he became PM, not be forced into recalling the diplomat early.
He also took issue with Mr Trump for saying the prime minister had failed to listen to his advice and been made to look "foolish" over Brexit.
"His comments about Theresa May were unacceptable and I don't think he should have made them," he said, remarks which prompted audience applause.
Mr Johnson said the US president had been "dragged into a British political debate" not of his making, but did suggest his outburst on Twitter - in which he called Sir Kim a "pompous fool" - had "not necessarily been the right thing to do".
While civil servants must be able to give confidential advice, he declined to comment on Sir Kim's future, only asking Mr Hunt to rule out "extending his term out of sympathy".
Tax pledge a 'mistake'
Both men have been criticised for making uncosted spending promises and offers of tax cuts during the campaign.
Mr Hunt sought to make capital out of Mr Johnson's pledge to give a tax cut to higher earners by raising the threshold at which people pay 40% tax from £50,000 to £80,000.
"It was a mistake, tax cuts for the rich," he said. "I have spent my life trying to persuade people that we are not the party of the rich."
Mr Johnson defended what he said was a "package" of measures to reduce the tax burden for both low and middle earners and which he said would boost the economy.
The show, entitled Britain's Next Prime Minister: The ITV Debate, was hosted by journalist Julie Etchingham in front of a studio audience of 200 people at MediaCityUK in Salford.
It took place as 160,000 or so party members get the chance to vote by post on who should succeed Theresa May.
The winner and next PM will be revealed on 23 July - it will be the first time a sitting prime minister has been chosen by party members.