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.
London Mayor Boris Johnson has said he would not quit as an MP if a Conservative government approved a third runway at Heathrow.