Billy Bob Thornton Says He Never Banged Amber!!!

Johnny Depp could have saved the paint -- and the blood --- when he allegedly wrote the words "Billy Bob" with what was left of his finger on a mirror ... because Mr. Thornton says he never hooked up with Amber Heard.

TMZ broke the story ... Amber and her people claim in March, 2015 Johnny accidentally cut his fingertip off in the middle of an argument with Amber ... and did his artful handiwork on the mirror after accusing her of cheating with Billy Bob when they co-starred in the flick, "London Fields," which was set to be released in 2015.

Billy Bob tells us the cheating allegation is ridiculous ... "completely false."

Thornton says he never socialized with Amber off set, other than a cast dinner. Even on the set, he says he didn't hang with her ... he was mostly around members of the crew. He adds he didn't even have a friendship with the actress ... it was all professional, on the set, and that was it.

As for Johnny, even though they were on the same set, Billy Bob says they had no interaction ... did not even speak.

Billy Bob's been married to wife Connie Angland since 2014. He says he's talked to Connie about the story and assured her it was Johnny's fantasy.