Speaking during a leadership hustings in Birmingham in the race to win the Tory crown and be next Prime Minister, Mr Johnson said: "I don't think they want to hear about that kind of thing."
The hustings event came a day after it emerged that officers were called to the London home Mr Johnson shares with partner Carrie Symonds after neighbours said there had been a loud altercation involving screaming, shouting and banging.
When asked by hustings moderator Iain Dale whether a person's private life has any bearing on someone's ability to discharge the office of Prime Minister, the crowd booed and Mr Johnson said: "Don't boo the great man."
Mr Johnson added: "I've tried to give my answer pretty exhaustively.
"I think what people want to know is whether I have the determination and the courage to deliver on the commitments that I'm making and it will need a lot of grit right now."
Mr Johnson said: "People are entitled to ask about me and my determination, my character and what I want to do for the country.
"Let me just tell you that when I make a promise in politics, about what I'm going to do, I keep that promise and I deliver."
Mr Dale told Mr Johnson he was "completely avoiding" the question.
Meanwhile Tom Penn, the neighbour who reported the incident, said in a statement reported by The Guardian that he recorded the altercation within his own home.
"After a loud scream and banging, followed by silence, I ran upstairs, and with my wife agreed that we should check on our neighbours.
"I knocked three times at their front door, but there was no response. I went back upstairs into my flat, and we agreed that we should call the police.
"The police arrived within five minutes. Our call was made anonymously and no names were given to the police. They subsequently called back to thank us for reporting, and to let us know that nobody was harmed.
"To be clear, the recordings were of the noise within my own home. My sole concern up until this point was the welfare and safety of our neighbours. I hope that anybody would have done the same thing," he said.
Mr Penn also defended his decision to reveal details of the incident to the newspaper, saying: "Once clear that no-one was harmed, I contacted the Guardian, as I felt it was of important public interest.
"I believe it is reasonable for someone who is likely to become our next prime minister to be held accountable for all of their words, actions and behaviours.
"I, along with a lot of my neighbours all across London, voted to remain within the EU. That is the extent of my involvement in politics.
"The unpleasant things being said about myself and my partner, and some quite frankly bizarre and fictitious allegations, have been upsetting for not only us, but also for family, friends and fellow Camberwell neighbours, who are currently being harangued by the media.
"I would ask that you leave private citizens alone and focus instead on those who have chosen to run for power within the public eye."