The suspension matches the fourth-longest in NHL history for an on-ice violation and will cost May almost $120,000 in salary.
Heinze needed nine stitches on his nose but was not seriously hurt. May received a match penalty for attempt to injure in the Saturday night game and has sat out the two games since the infraction.
May met with Colin Campbell, the league's director of operations, in Toronto on Tuesday.
"I have to be held accountable," he said before his hearing with Campbell. "I've never gone after anybody like that before in my career."
"Honestly, I just wanted to tap him on the arm," May told the Arizona Republic.
He apologized to Heinze after the game. Heinze said he accepted the apology.
"I told the trainers, 'Stitch me up quickly, boys. I'm on a power play,'" he told the newspaper.
Campbell said that while May "has not previously been suspended by the league for a supplementary discipline incident, he must be held accountable for the reckless use of his stick in this instance.
"It is fortunate Mr. Heinze did not suffer a more serious injury. The message to all players in the NHL should be clear: the act of a player striking an opponent in the head with his stick will carry severe consequences."
May will be eligible to return Jan. 4 against the New York Rangers.
"I think it was a very harsh suspension. We're going to miss him terribly over the next 18 games," said Coyotes general manager Bobby Smith, adding that the team made its case to the NHL on Tuesday and has no avenue for an appeal.
"I certainly hope the (Marty) McSorley incident had nothing to do with this," Smith said. "I think it had to do with the fact that the league decided this year to lower the bar on this type of incident."
Blue Jackets general manager Doug MacLean told The Fan, an all-sports radio station in Toronto, that the stiff penalty was necessary.
"Careless with the stick is a problem, there's no doubt about that," MacLean said. "Steve Heinze only got nine stitches on the end of the nose, but it very well could have been an eye or it could have been a broken jaw or is could have been something more serious.
"I guess it comes to a point that it's unfortunate for Brad May, but maybe it's good for everybody to see how severely this is going to be treated."
Under the terms of the Collective Bargaining Agreement, May will forfeit $117,647 of his salary. The money goes to the Players Emergency Assistance Fund.
McSorley is currently sitting out one year for hitting Donald Brashear in the head with his stick Feb. 21. Gordie Dwyer of Tampa Bay was suspended for 23 games for abusing officials and coming out of the penalty box to fight in an exhibition game against the Washington Capitals last Sept. 19.
Dale Hunter of the Capitals sat out 21 games for hitting Pierre Turgeon, who was celebrating a goal, during the 1993 playoffs. Tom Lysiak of the Chicago Blackhawks was suspended for 20 games for tripping an official in 1983.
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