Heeney will miss the Swans' clash with North Melbourne on Saturday, while the ban makes him ineligible for the AFL's highest individual honour this season.
The 28-year-old was initially suspended by the match review officer for striking St Kilda defender Jimmy 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.
Sydney appealed on three grounds, which were all dismissed at the end of a two-hour hearing.
The Swans argued the tribunal 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.
Sydney also cited the "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.
They argued Heeney's record - two fines and no suspensions in 193 games - was no different to Cameron's.
But it was to no avail, with the ban left to stand.
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."
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