The Patriots were on their bye week at the time.
"I feel bad, personally. I feel bad," Gronkowski said during his first media availability since the video was posted. "I talked to [the fan], and he assured me that he took no offense in any way. I know others did, and I apologize to those who took offense."
Gronkowski referred to the fan only as Joe and said he is a "good guy and a friend."
When a reporter asked another question about the video, Gronkowski said after addressing it once his focus would be on the team's game Monday night against the Carolina Panthers.
Gronkowski was heard on a tape posted by TMZ Sports mocking a man wearing Gronkowski's No. 87 jersey.
"They told me he could only cook fried rice," Gronkowski said into a microphone at the club. He then called the man "Leslie Chow," the Asian character played by Ken Jeong in "The Hangover" movies.
On the video, a person involved with the event apparently jokingly reminded those in attendance that cellphone video was to be left with organizers.