And he has said he would be prepared to keep no deal on the table in any future negotiations.
He warned that failure to deliver on the referendum result would create an "existential threat" for both Labour and the Conservatives.
"Around the country there is a mood of disillusion, even despair, at our inability to get things done", he said.
At his launch event, Mr Johnson had to fend off a series of reporters' questions about his past character and record in office.
But the plan did not garner enough support to pass the motion and it was voted down by a majority of 11 - with MPs voting 309 to 298 against it.
He acknowledged that his use of language - such as his description of Muslim women who wore the burka as letter boxes - sometimes resulted in "some plaster coming off the ceiling".
Referring to his record as mayor of London, he said: "I do what I promise to do as a politician".
Labour used an opposition day to table a controversial business motion that sought to "put power back in the hands of Parliament" and pave the way for another vote in June that, if passed, would block a no deal.
Sajid Javid, one of the candidates vying to succeed British Prime Minister Theresa May, said his rival Boris Johnson was "yesterday's news" and that it was time for a new generation to take control of the Conservative Party.
Although the defeat closes off one route for parliamentarians to try and prevent a no-deal exit, the flexibility of Britain's parliamentary system means others could be found in the weeks and months ahead.
The proposal was put forward after two of the candidates in the running to be the next Prime Minister, Dominic Raab and Esther McVey, both suggesting they would prorogue Parliament to stop MPs blocking leaving the European Union without a deal.
She told BBC Radio 4's Today programme: "This is about that safety valve, that lock in the process, so that somebody who may find themselves elected a leader of the Tory Party on a promise of, in Dominic Raab's case, proroguing Parliament, and locking Parliament out of this process - they can't do that".
Shadow Brexit minister Jenny Chapman said the vote on blocking a no-deal Brexit was a "safety valve" aimed at Tory leadership contenders trying to freeze MPs out of the decision.
"They would have to come back to Parliament and get the consent of MPs".