Kimmel said he would accept an apology from Rodgers, but he didn't expect one.
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"A decent person would apologize," Kimmel said Monday during his first show of the new year. "But he probably won't."
Kimmel laced into Rodgers in his monologue, calling him "hamster-brained" and said that he got two 'A's' on his report card - "they're both in the name Aaron."
"It might be time to revisit that concussion profile, Aaron," Kimmel said.
Rodgers insinuated Kimmel's name might appear in a list that that identifies associates of Epstein, a convicted sex offender who was accused of sex trafficking involving underage victims before he died by suicide.
Kimmel has denied having any association with Epstein and said Rodgers' comments put his family in danger.
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"Keep it up and we will debate the facts further in court," Kimmel posted last week on X, formerly known as Twitter.
Rodgers addressed the comments he made about Kimmel Tuesday when he appeared on the "Pat McAfee Show."
The New York Jets quarterback denied he implied Kimmel was a pedophile and condemned those who do, but he stopped short of apologizing for his role in escalating their burgeoning feud.
The Associated Press contributed to this report.