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The Dutch were knocked out at the last-16 stage after losing 2-0 to the Czech Republic on Sunday.
In a statement on the Dutch federation's website, De Boer said: "I have decided not to continue as national coach. The objective has not been achieved, that is clear.”
De Boer took charge of Holland last September after Ronald Koeman stepped down to take the Barcelona job.
Holland were impressive in the group stage at Euro 2020, winning all three matches against Ukraine, Austria and North Macedonia.
But they lost against the Czechs after the dismissal of defender Matthijs De Ligt.
De Boer said: "When I was approached to become national coach in 2020, I thought it was an honour and a challenge, but I was also aware of the pressure that would come upon me from the moment I was appointed.
“That pressure is only increasing now, and that is not a healthy situation for me, nor for Dutch football on its way to World Cup qualification.
“I want to thank everyone, of course the fans and the players.”
De Boer’s departure from the Holland job marks the end of another disappointing chapter in his coaching career, which started well at Ajax but has since featured short-lived spells at Inter Milan and Crystal Palace.