Shortly before lunch on Sunday's final day, Bairstow - the last of England's recognised batsmen - fell for 10 in bizarre fashion when he wandered out of his ground after ducking a Cameron Green bouncer and quick-thinking wicketkeeper Alex Carey under-armed the ball onto the stumps.
Bairstow thought he had secured his ground by tapping his bat behind the crease.
Australia captain Pat Cummins could have withdrawn the appeal but the decision was referred to third umpire Marais Erasmus, who ruled Bairstow had been stumped, with England now 193-6 chasing a target of 371.
As spectators jeered the decision and taunted Australia with chants of "Same old Aussies, always cheating", an angry Stokes then on 62, launched a blistering assault on the bowling.
The all-rounder made a stunning 155 but his brilliant innings could not prevent England falling to a 43-run defeat as Australia went 2-0 up in the five-match series.
Bairstow had been dismissed off the last ball of the over, with Stokes telling the BBC: "When is it justified that the umpires have called over?
"Is the onfield umpires making movement, is that enough to call over? I'm not sure. Jonny was in his crease then out of his crease.
"I am not disputing the fact it is out because it is out."
However, he added: "If the shoe was on the other foot I would have put more pressure on the umpires and asked whether they had called over and had a deep think about the whole spirit of the game and would I want to do something like that.
"For Australia, it was the match-winning moment.
"Would I want to win a game in that manner? The answer for me is no."
Stokes insisted his blizzard of boundaries - he struck nine fours and nine sixes in total - had not been motivated by anger at the manner of Bairstow's exit but rather the situation of the match.
"My innings wasn't a response to the dismissal, it was a response to where the game was," he said.
"I had to be mindful and strategic about the risk I was going to take and felt my best option was taking the fielders out of play."
'Not sneaky'
Cummins, meanwhile, insisted he had no regrets over the manner of Bairstow's dismissal.
"There was nothing untoward or sneaky," said Cummins, who led Australia's attack with 3-69.
"Jonny was leaving his crease every ball. He did it four or five balls. You're meant to stay in your crease in cricket. Bairstow has tried it with a lot of our guys."
Cummins, asked if there would now be "bad blood" between the sides.
"Not from our team, you might have to ask them (England)," the fast bowler replied.
Cummins, bidding to become the first Australia captain to win an Ashes in England in 22 years, added: "It was a wonderful contest. It's been a great Ashes series so far."
