Vermette was suspended by the NHL on Feb. 16 for 10 games for slashing a linesman, and commissioner Gary Bettman upheld the suspension on Feb. 25.
Vermette and the union had seven days to decide whether to appeal Bettman's decision to an independent arbitrator but decided not to go that route.
He will lose $97,222 in salary. He has served six games of the suspension so far and is eligible to return at home March 12 against the Washington Capitals.
The normally mild-mannered Vermette appeared to act out of frustration when Alphonso dropped the puck before the forward had put his stick in place on the ice. Officials immediately assessed a game misconduct to Vermette.
In the appeal, the NHLPA sought a reduction to a five-game suspension, noting Vermette's usually exemplary behavior. He has been whistled for only two major penalties in a 13-season career.
After the appeal was rejected, Vermette released a statement acknowledging his actions warranted a suspension but saying he felt a 10-game ban was "excessive."
"First and foremost, I have great respect for the officials and the integrity of the game," Vermette said in the statement. "Also, I agree that I deserve a suspension. I should never make contact with an official. At the earliest possible opportunity, I apologized to Mr. Shandor Alphonso. I was trying to get his attention and was not trying to hurt him. However, using my stick was a mistake, and I accept full responsibility for my actions.
"I certainly hope my track record in the NHL will earn me the benefit of the doubt regarding this incident. I feel a 10-game suspension is excessive and will be reviewing my options. I look forward to returning to the lineup as soon as possible."
ESPN's Pierre LeBrun contributed to this report.