Mr Johnson, who returned to the Commons at the election, told LBC he would be best placed to continue to fight expansion by staying in Parliament.
The Conservative manifesto said the party would "take account" of the Davies Commission into runway options.
The report, which has two Heathrow and one Gatwick options, is due next month.
Mr Johnson has been an outspoken opponent of further expansion at Heathrow.
He told LBC that if it did go ahead there would have to be a new congestion charge zone around it to tackle the traffic and pollution caused.
Fellow Conservative MP Zac Goldsmith has pledged to quit the party and force a by-election if the Conservatives - who scrapped Labour plans for a third runway in 2010 - complete a U-turn on the issue of Heathrow expansion.
'Filed vertically'
In his acceptance speech on being elected MP for Uxbridge, Mr Johnson said he would lie down "in front of those bulldozers and stop the building, stop the construction of that third runway" at Heathrow.
Asked on his regular radio phone-in show if he would resign, Mr Johnson pointed out he did not have a government job to resign from - on Monday he was named as someone who would attend the Conservatives' political cabinet in Downing Street while concentrating on his final year as London mayor and being an MP.
He was then asked by Nick Ferrari if he would force a by-election.
Mr Johnson said he thought he "would be best off staying in Parliament to fight the case".
He added that he thought Heathrow was "undeliverable" and that if the Davies Commission did choose a new runway there as its preferred option, he hoped it would be "filed vertically [shelved]" as a series of other reviews in past decades had been.

An IFA Disciplinary Committee opted not to impose any sanction after finding that Carrick did not properly implement manager Gary Haveron's touchline ban.
The committee said it would "exercise its discretion" in the matter.
A Warrenpoint Town statement argued that the decision "contradicted the IFA's obligation to apply objectivity".
"The board of Warrenpoint Town have read the IFA Disciplinary Committee determination regarding the breach of Article 23.1 of the IFA Disciplinary Code by the Carrick Rangers FC Manager," the statement began.
"We are shocked that the IFA having found Carrick guilty failed to implement the requisite admonishment as per established rules.
"The board will be meeting soon to discuss further this matter and the complaint we formally raised surrounding eligibility which is in abeyance."
Carrick faced a possible three-point deduction and relegation from the top flight, plus a possible fine of at least £350, if the outcome of the hearing did not go in their favour.
However in a statement released on Wednesday night, the committee indicated that "the interests of justice were best served by not imposing a sanction in all the circumstances".
As it stands, Ballinamallard United and Institute await a date for the second leg of their promotion-relegation play-off, the Mallards having won the first leg 2-1 at the Riverside Stadium on 6 May.
Haveron sat out a three-game ban handed out by the IFA, but was in the dugout for his club's match against Dungannon Swifts on 23 April when he should not have been.
"The club did not explain to the Committee's satisfaction the reasons why the start date for the suspension set out in the initial charge letter (18 April 2016)  was not complied with and therefore the challenge from Carrick Rangers was not upheld," read the IFA statement released on Wednesday night.
"The Committee took into account the points made on behalf of the club and decided to exercise its discretion under the overriding objective as outlined in Articles 1.6 and 1.7 of the Disciplinary Code and found that no sanction should be applied."
"In applying its discretion it took into account, what it found to be the honest intention of the club to serve a suspension, the fact that three matches had been served and points made regarding possible confusion in the way in which the suspension was to be implemented."
London Mayor Boris Johnson has said he would not quit as an MP if a Conservative government approved a third runway at Heathrow.
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Warrenpoint Town say they are "shocked" by an IFA ruling that confirmed their relegation from the Premiership and saw Carrick Rangers stay in the top flight.