As Brown left the stadium, a Gers supporter asked the former Scotland international, "How's your sister?" in reference to the player's sibling Fiona, who died from skin cancer aged just 21.
Footage of the incident posted on social media shows Brown staring down the individual before getting on the Celtic team bus.
In the clip, a police officer can be heard reprimanding the fan while a fellow supporter adds: "Come on to **** man, that's out of order".
The Sun reports that Brown was given a police escort to a local golf course after the match, which Celtic won 2-0 thanks to goals from Odsonne Edouard and Jonny Hayes.
Fiona had been ill for around 12 months before passing away in 2008 - just weeks after the death of Celtic legend Tommy Burns from the same disease.
Supporters of both Celtic and Rangers have united to condemn the incident, with fans branding it "sick", "tasteless" and "vile".