The organization also want the NHL to educate its fans about such hate speech.
Video replay appeared to catch Simmonds making an anti-gay slur to Avery during a preseason game in Philadelphia on Monday night. Avery confirmed that Simmonds made the remark, and Simmonds didn't deny saying it.
"Hate speech and anti-gay slurs have no place on the ice rink," GLAAD acting president Mike Thompson said in a statement. "The word that Simmonds used is the same word that is hurled at LGBT youth on the playground and in our schools, creating a climate of intolerance and hostility.
"He should not only apologize for this anti-gay outburst, but the Philadelphia Flyers and the NHL have a responsibility to take action and educate their fans about why this word is unacceptable."
GLAAD said it is talking to the Flyers and the NHL about what specific steps can be taken.
The NHL didn't immediately comment.
GLAAD said it has worked with sports leagues such as the NBA, Major League Baseball, and the World Wrestling Federation to address issues of homophobia in sports.
