The AFL announced the ban on Sunday after finding Graham guilty of conduct unbecoming over the incident during the July 4 match won by GWS at Optus Stadium.
It said the incident was self-reported by Graham to his club, who reported it to the AFL on Friday morning, prompting an AFL integrity unit investigation."Graham used a highly offensive homophobic slur against a GWS Giants opponent, which demeans and denigrates persons regardless of their sexuality," the AFL said in a statement.
It said it took into account Graham's self-reporting of the incident, along with his private apology and contrition. Graham also rang the AFL CEO directly to apologise.
A two-time premiership player with Richmond before joining the Eagles, 27-year-old Graham will not be eligible to play for West Coast for the next four weeks and he will undertake Pride in Sport education.
"I am very remorseful for the word I used and know such language has no place in our community or our game," Graham said in a club statement.
"I have offered my unreserved apology to the Giants player and Greater Western Sydney and would also like to publicly apologise for my comment.
"What I said does not reflect my values or our club values. I made a mistake and need to be better and will prove that through my actions going forward."
West Coast CEO Don Pyke said Graham had been honest in addressing the matter and understood its harmful effects.
"Player welfare was front of mind throughout the process and we were in regular contact with the Giants following the match and earlier this week around this situation," he added.
"With no match day report being lodged, both clubs agreed the appropriate course of action was to refer the matter to the AFL."
AFL head of integrity and security Tony Keane said: "The AFL is very clear that homophobia has no place in our game, nor in the community.Â
"We have come a long way as a code, but this incident demonstrates there is so much more to be done, and we will continue to work together to improve our game as a safe and inclusive environment for all.
"Graham's sanction would have been higher if he hadn't self-reported and understood his comment was totally inappropriate."
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