The Northern Ireland international is seen playing with his private parts and singing as he makes made his way down a road, in the video which emerged after Saturday’s 2-2 draw with Fleetwood.
Manager Karl Robinson said that the club will give him a strict warning for what he described as “a moment of madness”.
“It is totally out of character,” said Robinson.
“Gavin is a quiet, respectful young man who is totally devastated by this, I have rarely heard anyone so remorseful for something that he regrets and knows he shouldn’t have done.
“Clearly he had drunk too much, and we will deal with that too, but whether you are a footballer or not that sort of behaviour is wrong on every level.
“We will sit down with Gavin today and we will remind him in no uncertain terms of his responsibilities.
“We have psychological and behavioural support here to help him get through this. He is a great kid and I hope the fans can forgive him for one moment of madness.”
A club statement read: “Oxford will today meet with Gavin Whyte and warn him as to his future conduct after an incident back home in Northern Ireland at the weekend.
“Gavin was filmed behaving in a way that is clearly unacceptable from a professional footballer and we will work with both Gavin and the Irish FA to ensure that he is aware of his responsibilities.”