Tortorella, who is not permitted to have any interaction with his team during the ban, tried to barge into the Calgary Flames locker room after the first period of last Saturday's game, which began with a 10-player brawl.
"Mr. Tortorella's actions in attempting to enter the Calgary Flames locker room after the first period were both dangerous and an embarrassment to the League," said NHL Senior Executive Vice President of Hockey Operations Colin Campbell.
Calgary coach Bob Hartley used a starting lineup of mostly fourth-line enforcers and Tortorella, who as the home coach was able to respond with whatever lineup he wanted, opted to send out his enforcers as well.
All 10 skaters engaged in a brawl when the game began and Tortorella could be seen screaming at Hartley during the melee. The fights resulted in eight ejections and 152 penalty minutes.
Hartley was fined $25,000 for his role in the in the brawl, the NHL said in a separate statement.
When the first period ended, TV cameras caught Tortorella involved in an altercation outside the Calgary locker room as he tried to get inside and confront Hartley.
"We respect the decision made by the National Hockey League today to suspend John Tortorella for 15 days from Sunday, with no contact with the team for six games," Canucks General Manager Michael Gillis said in a statement.