The 53-year-old, widely tipped to quit after the tournament, said: "As a proud Englishman, it has been the honour of my life to play for and to manage [England].
"It has meant everything to me and I have given it my all. But it's time for change, and for a new chapter."
The race to appoint his successor has started with fans' favourite and former Liverpool boss Jurgen Klopp, Newcastle manager Eddie Howe and former Brighton coach Graham Potter among the frontrunners for "the impossible job".
Prince William, 42, president of the Football Association, who watched Sunday night's 2-1 defeat to Spain with his son Prince George, 10, said: "Gareth, I want to thank you not as the president of the FA, but as an England fan. Thank you for creating a team that stands shoulder-to-shoulder with the world's finest in 2024.
"Thank you for showing humility, compassion and true leadership under the most intense pressure and scrutiny.
"And thank you for being an all-round class act. You should be incredibly proud of what you've achieved."
And last night captain Harry Kane tweeted: "Boss. Thank you for everything you have done for me and our country! You have been one of England's greatest ever managers and it has been a pleasure to have served as your captain throughout.
You helped our nation connect and believe in our team again.
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