WASHINGTON (CNN) -- Former Tennessee Sen. Fred Thompson and filmmaker Michael Moore are going at it online over Moore's recent trip to Cuba for his health-care documentary "Sicko."

Thompson, currently an actor on the popular "Law & Order" television show and a potential GOP presidential candidate, accused Moore in a May 2 post on the conservative Web site National Review Online of parroting Fidel Castro's "story of free, high quality Cuban health care." Thompson called reports of high quality health care in Cuba a "myth."

In a letter responding to Thompson and provided to CNN Tuesday, Moore challenges the Tennessee Republican to a healthcare debate and, citing reports that Thompson openly enjoys Cuban cigars, accused him of hypocrisy.

The filmmaker suggested the winner of the debate could be decided by an American Idol-style vote.

But, citing scheduling concerns, Thompson Tuesday declined the invitation in an online video as he chewed on a cigar, and suggested mental institutions are something Moore "oughta think about."