Jeremy Hunt has urged Tory leadership rival Boris Johnson not to be “a coward” about facing public scrutiny.
Mr Hunt said he was “not interested” in his private life but he should “man up” and debate with him on TV this week.
Meanwhile, Mr Johnson has warned the UK will face a “democratic explosion” if it does not leave the EU by 31 October.
But Mr Hunt challenged him to reveal whether he would call a general election if MPs refused to allow the UK to leave without a deal on that date.
After Prime Minister Theresa May failed to get her Brexit deal through Parliament earlier this year, the date of the UK’s departure for the EU was moved to 31 October.
Mr Johnson is under pressure to answer questions about a row with his partner in the early hours of Friday which led to police being called to his London home.
The Metropolitan Police has said it will not be taking any further action over the incident and his supporters have rallied around him.
Former International Development Secretary Priti Patel told BBC Radio 4’s Today programme a recording of the argument, given to the Guardian newspaper, was part of a “politically-motivated series of attacks”.
“That is not the type of behaviour that you’d expect in our country, that’s the type of behaviour associated with the old Eastern bloc,” she added.
Mr Johnson refused to answer questions about the incident at a hustings event on Saturday, instead insisting his stance on Brexit was what mattered to the public and to the Conservative Party members who will choose the next leader.
In his Daily Telegraph column on Monday, he said of the 31 October deadline: “This time we are not going to bottle it. We are not going to fail.”
He said it was “disgraceful” the UK was still in the EU three years after it voted to leave, and exiting the EU would “renew the national faith in democracy”.
He did not address questions about his private life in the column.
Writing in the Times, Mr Hunt called for a “fair and open contest, not one that one side is trying to rig to avoid scrutiny”.
“Only then can you walk through the front door of No 10 with your head held high instead of slinking through the back door, which is what Boris appears to want.”
He told BBC Radio 4’s Today programme it was “very disrespectful” of Mr Johnson to refuse to do “any tough media interviews” and urged him to take part in a Sky News leadership debate scheduled for Tuesday.
The two men are due to face off on ITV in July, but by then voting papers will already have been sent to party members.
Mr Hunt said he feared a government led by Mr Johnson would rapidly collapse, because he would be unable to hold together a coalition of supporters that range from MPs who back no deal to others who feel it would be totally unacceptable.
“If you are not clear about exactly what you are going to do, that coalition will collapse immediately and you will have Corbyn in Number 10,” the foreign secretary said.
He said Mr Johnson must explain how he could guarantee the UK would leave the EU on 31 October if Parliament voted to stop a no-deal Brexit, as it did in a non-binding vote in March.
Mr Hunt ruled out calling a general election in such a circumstance – saying it would destroy the Conservative Party – and demanded that Mr Johnson be clear whether he would do the same.
Health Secretary Matt Hancock, who withdrew from the leadership contest after coming sixth in the first ballot of the party’s MPs, told BBC Breakfast Mr Johnson had the “best chance” of securing a new Brexit deal with the EU.
Mr Hancock said it was “total nonsense” to suggest Mr Johnson was not open to scrutiny, drawing attention to the various hustings he has taken part in.
“He’s got the energy, he’s got the support from right across the party, and I think that’s why he’s the right man for the job,” Mr Hancock added.
‘No confidence vote’
In a separate development, defence minister Tobias Ellwood told the BBC’s Panorama programme that “a dozen or so” Conservative MPs would support a vote of no confidence in the government to stop a no-deal Brexit.
A no-deal exit would see the UK leave the customs union and single market overnight and start trading with the EU on World Trade Organisation rules.
Opponents say it would cause huge disruption at the borders and be catastrophic to many firms reliant on trade with the continent.
Next month around 160,000 Conservative Party members will choose the next leader of the Tory Party – and the next prime minister.
Members will receive their ballots between 6 and 8 July, with the new leader expected to be announced in the week beginning 22 July.