Police Lt. Robert Swanigan said the altercation at the Georgetown clinic began Monday with a verbal exchange between Ed Wilson, a 68-year-old Kentucky fan, and Charles Taylor, a 71-year-old Louisville fan.
The men were arguing about who will win when the teams meet Saturday in the Final Four.
"I think this is a first at a dialysis center," Swanigan told WKYT-TV.
Swanigan says Wilson, who already was hooked up to the dialysis machine, flipped off Taylor, who was waiting his turn.
That's when Taylor punched Wilson in the face.
"He just happened to think U of L would beat UK and he started to run his mouth," Wilson told WKYT. "That's what started it."
Taylor said he was actually talking about the matchup with someone else.
"He was meddling," Taylor said. "And told me to shut up and gave me the finger.
"I went up to him and I hit him. Didn't hit him that hard, but I hit him."
Swanigan said he thought tensions would rise as Saturday's game approaches, but the fight at the clinic was stranger than they expected.
Georgetown is about 13 miles north of Lexington, where the University of Kentucky is located.
Wilson declined to file charges, and Georgetown police say the case is closed.
Information from The Associated Press was used in this report.