The 36-year-old's contract with the club had another year to run.
The Rams have been in administration since September last year and were relegated to League One last season.
"I feel the club now needs to be led by someone with fresh energy and not affected by events that have happened over the last 18 months," Manchester United great Rooney said.
"My time at the club has been a rollercoaster of emotions, both highs and lows, but I have to say that I have enjoyed the challenge.
"I will remember my time at Derby with great pride and affection and would like to thank all my staff, players and of course the fans for their incredible support. I will never forget you and hope to see you all again in the near future and in happier times.
"Finally, I am aware that the club still have interested parties who wish to take over the running of the club. To them I say this, Derby County is a great club with a great history and great fans. I wish you all the best and much success for the future."
In a statement, Quantuma said: "The joint administrators are very disappointed that Wayne has taken the difficult decision to leave the club and we have spent some time today trying to persuade him to stay but understand his reasons for wanting to go.
"We are extremely grateful to him for his excellent work in the face of challenging on-field circumstances in the 2021-22 season and admire the manner in which he has led the team, the club and the local community through various off-field issues.
"The joint administrators recognise that staff and supporters will be frustrated and equally disappointed by this news, but we all wish him, his wife Coleen and their four boys every success in the future and we are sure they will always be welcomed back."