Greene, 20, watched the game while having a few drinks with mates.
Near the end of the match, Greene’s group allegedly became involved in a scuffle with a group of older patrons at the stadium's Locker Room bar.
It is understood a man in his 50s suffered a black eye and cut to the head as a result of the incident.
It is believed at least two members of Greene's group were ejected from the venue.
Greene’s exact role in the incident remains unclear.
A spokesman for Etihad Stadium confirmed there was an incident.
"There was an incident in the Locker Room and two people were ejected outside the Locker Room, and they were ejected by police and stadium security," Etihad Stadium spokesman Bill Lane confirmed to AFL.com.au.
It’s not clear whether CCTV captured anything related to the incident.
A spokesman for Greater Western Sydney declined to comment on the matter.
It's expected the club's leadership group and officials will meet in the coming days to discuss the situation.
Police were not in a position to provide comment on Wednesday night.
In the hours after the Etihad Stadium incident, Greene and his group ventured to Zagames, a licensed venue in Melbourne’s south-eastern suburbs.
There, it's alleged Greene and a 19-year-old man from East Malvern were involved in a confrontation with a bouncer.
Greene has been charged with intentionally causing serious injury, recklessly causing injury, affray, assault in company, assault by kicking, assault with a weapon, unlawful assault, making threats to inflict serious injury, being drunk in a public place and criminal damage.
Both men were granted bail.
The pair is due to appear in the Melbourne Magistrates’ Court on September 25.
Giants players are on leave ahead of the club's bye weekend. The club is yet to indicate what action of its own it intends to take.
Pies' big selection calls, Dees on Fritsch trade talk, key Hawk returns
Footy Feed: Sarah Olle with the latest news
03:00
Bombers equal debutants record as injury list grows
As its injury list continues to grow, Essendon has confirmed mid-season draftee Liam McMahon as its 13th debutant for 2025, equalling the record set by Fitzroy in 1991
02:40
Match Previews R19: Essendon v GWS
Sportsbet's Nathan Brown and Josh Jenkins preview the game between the Bombers and Giants at Marvel Stadium