Wicketkeeper Carey was drawn into a 'Spirit of Cricket' row after he stumped Bairstow, who had carelessly wandered out of his crease believing the ball was dead, during this month's second Test at Lord's.
Australia won that match by 43 runs despite Ben Stokes' stunning 155.
But the England captain' heroics were overshadowed by the fall-out from Bairstow's unusual exit.
British Prime Minister Rishi Sunak and Australian counterpart Anthony Albanese traded verbal bouncers after Bairstow's dismissal sparked a chorus of boos at normally sedate Lord's. Three MCC members were suspended for abusing visiting players in the pavilion.
There was, however, widespread agreement that Carey, according to the Laws of Cricket, had effected a legitimate dismissal.
With the series still in the balance after England's three-wicket win in the third Test at Headingley last week left Australia 2-1 ahead, Carey was adamant he'd have no doubts about doing something similar in either of the remaining matches at Old Trafford or The Oval.
"If there was an opportunity to get a stumping, I definitely would," he told reporters. "To see how much has played out since then it's been a little bit surprising.
"There's been some nasty stuff been said but it is the Ashes. There was nasty stuff said before that as well. I feel really well supported, I think the whole group does.
"From Australia, I still think we've got lots of fans and from England, I don't think we've made any, but we probably didn't lose any either."
Pretending to hold the ball and other forms of 'fake fielding' that might deceive a batsman are outlawed.
But Carey's reaction after Bairstow ducked a Cameron Green (24) bouncer was instantaneous. The batsman was in his crease when Carey threw the ball but out of it by the time the ball knocked off the bails.
"We were switched on to the fact that it was a bouncer plan and it felt like Jonny was pretty switched on to getting out of the way, he wasn't playing any shots," explained Carey.
"When he ducked, his first movement was pretty much out of his crease, so instinctively I grabbed the ball, threw the stumps down and the rest is history, as they say.
"As soon as I got it, I threw it straight away. Once the bail has come off, it's up to the third umpire to deem it out or not out, or the on-field umpires -- and it was given out."
