The Tory leadership candidate appeared to deliver a retort to his rival to be prime minister as he vowed "we are not going to bottle it" on the EU exit date of October 31.
But Jeremy Hunt called on Mr Johnson not to be a "coward" by avoiding a live TV debate with him this week, as he suggested otherwise he would be "slinking through the back door" of Number 10.
• READ MORE: Scottish Independence: Deputy PM warns Nicola Sturgeon against fresh campaignMr Johnson remains under pressure to explain why police were called to the home he shares with partner Carrie Symonds, but in his column for the Daily Telegraph he attempted to switch attention back to political rather than personal issues.
He wrote: "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," he said.
• READ MORE: Nicola Sturgeon 'absolutely confident' Scots are ready to vote for independenceMeanwhile, 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.
"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," he told BBC1's Panorama programme.
Foreign Secretary Mr Hunt has previously attacked Mr Johnson over reports he was "bottling" a live TV debate before postal ballots are returned.
Stepping up pressure on his rival, Mr Hunt insisted that while he has no interest in debating Mr Johnson's private life, he wants to challenge him on television over his commitment to taking the UK out of the EU by the end of October.
In an article in The Times, Mr Hunt said: "A new prime minister needs the legitimacy of having made his arguments publicly and having them subjected to 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 added: "Don't be a coward Boris, man up and show the nation you can cope with the intense scrutiny the most difficult job in the country will involve."
Mr Hunt said on Sunday that he thought any candidate for prime minister "should answer questions on everything".
Hunt supporter and International Trade Secretary Liam Fox signalled it was better for Mr Johnson to be clear about what had happened on Friday while shadow communities secretary Andrew Gwynne said Mr Johnson was "completely unsuitable" to be prime minister after the incident.
Mr Johnson repeatedly refused to explain to Tory grassroots at a Saturday hustings what went on at the south London flat in the early hours of Friday morning, saying the party faithful did not want "to hear about that kind of thing".
Mr Johnson's campaign for Number 10 was 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.