Footage of the incident showed Tui and the fan getting physical with each other - and at one point the player could be seen trying to grab the fan.
Police arrived after Tui's Wallaby teammates Jack Maddocks, Marika Koroibete and Samu Kerevi and a Rugby Australia staffer were forced to step in.
Eventually the fan was led away by the throat by another fan who was present during the scuffle.
The fan who began the verbal tirade allegedly brought children who were nearby to tears, according to News.com.au.
Wallaby player Reece Hodge was being interviewed as the scuffle started, according to Fox Sports, and he said: "It's very disappointing and sad to see our fans like this."
The Wallabies coach addressed the media following Tui's request to leave the season.
Cheika said: "It's one thing being upset with a team... [but] what went down there was the supporter pushed Lukhan's sister.
"Lukhan didn't like that, and that's fair enough."
He said following the death of Tui's stepfather what happened was inappropriate.
He said that while the fan probably didn't know who he was pushing "you shouldn't be pushing a kid in the crowd anyway".
Argentina won the match 23-19. It was Argentina's first win on Australian soil for 35 years and dropped Australia to seventh in the world rankings.