The highly-rated 19-year-old is Doncaster's starting left-back and his absence means that manager Paul Dickov will turn to Manchester City loanee Reece Wabara to fill in.
Rovers physio Alex Dalton told the club website: "James had his operation last week and everything went all right. It was as expected with no complications, so we should see him back in around 12 weeks."
Donny currently lie 17th in the Championship after 10 games of the season. They next travel to Reading on Saturday.