Rice was filmed approaching the official in the tunnel after the 1-0 second-leg defeat ended the Hammers' hopes of European glory.
Trailing 2-1 from the first leg, West Ham's task became more difficult when Aaron Cresswell was sent off for denying a goalscoring opportunity for Frankfurt forward Jens Petter Hauge in the 19th minute, before Rafael Santos Borre's goal sealed the Hammers' fate.
While there were few complaints about Cresswell's dismissal, Rice suggested 'every decision' went the home side's way in a post-match interview - and now the midfielder risks a UEFA investigation after his behaviour towards the referee in the tunnel was captured on film.
In footage obtained by Fox Sports Mexico, Rice shouts at Gil: "It's so poor, all night it's so bad. How can you be that bad?
"Honestly, you've probably been f****** paid. F****** corruption."
Speaking to BT Sport after the match, Rice admitted the Cresswell red card was not the source of his frustration.
"The sending off was probably a sending off but some of the other decisions were poor tonight," Rice said.
"The ref seemed to make every decision go for them. When you have someone against you, you can't do much."
Meanwhile, Hammers boss David Moyes, who was also sent off for kicking a ball back at a ball boy, echoed Rice's feelings about the officiating.
When asked about the decisions against West Ham on the night, the Scot responded: "Lots of complaints, yeah. Really disappointed. Things didn't go for us on the night. Small things didn't happen but I am really proud of the players."
He added: "We have enjoyed being in the competition. Not sure we have enjoyed the officiating. We have a pretty good team so I think we can try again. We will be a bit wiser and can hopefully do it more often."