Heeney will miss the Swans' clash with North Melbourne on Saturday, while the ban also makes him ineligible for the AFL's highest individual honour this season.
"I'm obviously extremely disappointed with that result," Heeney said in a statement soon after the appeals board's verdict was handed down on Thursday night.
"It's pretty shattering, to be honest.
"I can guarantee one thing - that when I'm back I'll make sure that I do this club proud, the supporters proud and my family proud."
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Heeney had been one of the leading Brownlow contenders before he was charged by the league's match review officer with striking Webster during Sydney's upset loss last Sunday.
The decision was upheld by the tribunal on Tuesday, prompting the Swans to take their case to the appeals board on Thursday night.
Sydney appealed on three grounds, arguing the tribunal had made an error of law, that no tribunal acting reasonably could have reached the decision it did and that there was manifest excessiveness in classifying Heeney's action as intentional.
All of the Swans' arguments were dismissed at the end of a two-hour hearing.
Heeney went to break away from Webster with a swatting action and caught his opponent across the face during Sunday's game at Marvel Stadium.
Webster was treated for a bloody nose on the field, but wasn't required to leave the ground and didn't need any ongoing medical attention.
Isaac Heeney (C) and his Swans teammates leave the field after their loss to St Kilda. (Joel Carrett/AAP PHOTOS)
Sydney made a submission against a rule tweak in January which declared similar actions would "usually" be graded as intentional, rather than careless.
On Thursday night, Swans legal counsel Duncan Miller insisted Heeney's action should have been graded careless, and the incident was in play rather than off the ball.
Miller suggested upholding the tribunal's decision would "wreak complete violence on the rest of the rules and guidelines."
"It would mean a lawful action which might accidentally result in a strike would somehow automatically be deemed to be intentional," Miller said.
AFL legal counsel Andrew Woods said Sydney had not made valid points as to why Heeney's suspension should be overturned.
Woods also acknowledged the Swans' assertion that Heeney did not intend for the contact with Webster to be high, but argued it was not a relevant point.
"It was a straightforward and unsurprising application of the guidelines," Woods said of the tribunal's decision.
Sydney also cited the contentious "good character" clause which helped the Brisbane Lions' Charlie Cameron overturn a one-match ban earlier this season, in part because of his clean record.
The Swans argued Heeney's record - two fines and no suspensions in 193 games - was no different to Cameron's, but the point was dismissed by the appeals board.
The appeals board deliberated for just 25 minutes before dismissing Sydney's arguments.
"To our mind, having regard to all the material that was before the tribunal, and looking carefully at the reasons expressed by the tribunal at the end of the hearing, we do not consider that the tribunal acted unreasonably or unjustly or unfairly," appeals board chair Will Houghton said.
Isaac Heeney was involved in this incident with Jimmy Webster late in the third quarter.— AFL (@AFL) #AFLSaintsSwans pic.twitter.com/jjOpkrVxtsJuly 7, 2024
Earlier, Sydney forward Hayden McLean joined coach John Longmire in defending Heeney, saying the swatting action his teammate used to break away from Webster is a common movement in the game.
"We sort of joked about it, like, you do that probably 40-50 times in a game," McLean said.
"You always try to get separation off your man, and it's just that unlucky thing.
"He was a bit sorry at the time, and I was behind him at the incident. He just literally said, 'Sorry, I didn't mean it.'
"You do that many times a game, and it's one of those things that just happened."
Heeney was named in Sydney's team to face North Melbourne before his appeal hearing.
His forced absence could open the door for veteran Luke Parker to play his first game of the season after injury and suspension.
Parker, Sam Wicks and ruckman Lachlan McAndrew were named as emergencies.
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