Burke, also Calgary's president of hockey operations, countered Tortorella's post-game comments after Sunday's game in Vancouver, where the Canucks won the regular-season finale 5-1.
Tortorella had objected to the actions of Flames head coach Bob Hartley when Canucks forward Daniel Sedin was hit from behind and driven into the boards by Flames forward Paul Byron late in the second period.
Sedin was taken off the ice on a stretcher after scoring twice in the game, while Byron was slapped with a five-minute major and a game misconduct.
"It's embarrassing to coach against the guy across from me tonight," Tortorella said. "Some of the things that went on when Danny was hurt, it's embarrassing.
"I don't like the way he does business. I don't like him, and eventually I guess why I am talking about it in this way is because I need to protect my players and a lot of people don't understand that, so I'll just leave it at that.
Tortorella feuded with Hartley in Vancouver on Jan. 18 when he tried to get into Calgary's locker-room following a line brawl. Tortorella was suspended for 15 days.
Hartley wouldn't bite Monday when reporters asked Calgary's head coach about Tortorella's comments, but Burke weighed in. He called Tortorella a friend before launching his counter-attack.
"Our coach acted completely appropriately last night and Torts oughta keep his mouth shut," Burke said. "Apparently what outraged him was the cardinal sin of our coach arguing for a penalty while Daniel Sedin was being tended to on the ice.
"Paul Byron hit him. We think it's a hockey play. We don't think it's worth five minutes. We don't think it's worth further review. Call a five-minute penalty because the player is hurt, that's fine with us.
"Now the player gets attacked, punched repeatedly, no penalty. Where are the minor penalties against our player? Smallest player on the ice besides Johnny Gaudreau gets mugged, nothing.
"Bob had the temerity and the audacity to argue with the officials for these penalties while Daniel Sedin was on the ice. No one likes to see a player get hurt. We hear Daniel is fine, which is great news, but that is when Bob has to argue that.
"What Bob Hartley did last night was exactly what John Tortorella would have done had it been one of our players lying on the ice and he felt there was an inequity in the penalty calls.
"John Tortorella was on my staff for the Olympics in 2010. He's a good guy. I don't know what's going on in his head, but you know what? Shut your mouth and worry about your team. Leave my coach alone."
The players in question seemed ready to make peace. Byron said he texted Sedin on Monday morning to apologize for the hit.
"He responded," Byron said. "I think he appreciated the text and I'm really glad he's OK."
Note to readers: This is a corrected story. A previous version had an incorrect position for Flames forward Paul Byron.