"My thoughts were that it was a very disrespectful comment and the league took action and I definitely agree with the way they took action today," Phaneuf said following a 3-1 loss to the Stars.
Flames winger Craig Conroy said Avery, his former Los Angeles Kings teammate, demonstrated the wrong sense of humour.
"He made stupid comments and it bit him," Conroy said.
"This organization will not tolerate such behaviour, especially from a member of our hockey team.
We hold our team to a higher standard and will continue to do so." Dallas concludes a two-game Alberta swing in Edmonton on Wednesday, but Avery will not be with the team.
Veteran forward Modano spoke out last month after watching Avery and Steve Ott become a sideshow during a game in Boston.