The 36-year-old Manchester United legend quit last week.
Writing on Twitter, Rooney said: "A massive thank you to all the fans, players, and staff at @dcfcofficial for your support over the last three years.
"This has been one of the hardest decisions I've had to make in my career.
"Thanks for taking me in as one of your own. Please get behind the next manager and give them what you gave me, trust, loyalty, and unbelievable support.
"Some great memories I'll never forget."
Rooney first joined Derby as a player-coach in January 2020 and, after spending time as interim boss, was named permanent manager in January 2021.