The pair were involved in an altercation earlier this month at a bar in Birmingham, where Warner is said to have punched the England opener.
However, Root has revealed that having received an apology, he is now keen to put the incident behind him.
"It's all done now. I hope we can move on," said the 22-year-old. "David Warner has apologised. I've accepted it. As far as I'm concerned, that's all behind us."
The Ashes between England and Australia gets underway at Trent Bridge on Wednesday, July 10.