Wigglesworth appeared to be targeted by a member of the crowd as he walked off the pitch after the final whistle, with an empty drinks bottle being thrown in his direction.
The incident capped off a chaotic encounter that saw five players sin-binned and another two shown red cards, and RFU chief executive Ian Ritchie admitted that it was not good advert for the sport.
"The RFU are looking at it, looking at the video and what went on. We'll make some views on it in a disciplinary fashion," Ritchie is quoted as saying by PA.
"I don't think anybody could look at it and say it was the greatest advert for the game, both in terms of the things that happened off and on the pitch. We'll be looking at it, but we have an independent disciplinary process.
"It will go through the right independent judicial process. I do not interfere with what happens in that process."
Bath won the match 18-17 after Wigglesworth had awarded them a 79th-minute penalty try against a Gloucester side that finished the game with just 11 men on the field.