Laughter is heard in the beginning of the video, which opens with a shot of Christie in close proximity to the fan's face.
"You're a big shot," Christie tells the fan as he walks away with a container of nachos.
"Appreciate that," the fan replies.
The fan, who was identified by ABC Milwaukee affiliate WISN as Brad Joseph, said the confrontation started when he yelled Christie's name as he was walking up the steps at Miller Park in Milwaukee.
"He was already quite a bit past me, and 30 feet away, I yelled his name and told him that he sucked," Joseph told WISN.
He told the station, "I called him a hypocrite because I thought it needed to be said. He then turned around and walked all the way back toward me and got up in my face for what seemed like a long time but was probably only about 30 seconds."
Christie then yelled, "Why don't you have another beer?" Joseph said.
"Which I thought was a decent comeback, and I thought that was was kind of funny," he said. "Then he started calling me a tough guy."
Joseph said he felt the right to express his opinion, since Christie is a public official.
"This is America, and I think we have the right to say what you believe, as long as it's not crude or profane," Joseph said.
According to WISN, Joseph is a friend of WISN reporter Ben Hutchinson, who began recording the video at the end of the interaction and posted it to his Twitter account.
This is Christie's second negative run-in with fans at an major league baseball game. On July 18, he was booed at Citi Field in New York after he caught a foul ball, The Associated Press reported.
His son Andrew Christie has worked for the Milwaukee Brewers since March, according to his LinkedIn account.