Denmark scraped through Group C in Germany with three draws, before losing 2-0 to hosts Germany in a round-of-16 tie featuring several controversial VAR interventions.
It was Denmark's second underwhelming tournament in a row, after they crashed out in the group stage at the 2022 World Cup in Qatar.
Hjulmand took the reins in 2020 after Age Hareide's departure, with his first tournament his most successful as Denmark reached the Euro 2020 semi-finals, losing to England in extra time.
"It has been an incredible honour and privilege to be allowed to coach our national team for four years," Hjulmand said in a statement released on Friday.
"I have given everything I had to achieve success and to get the people to rally around the team.
"Representing your country is the absolute greatest thing I can imagine, and I will look back on a fantastic time with great support for our dream.
"When I look two years ahead to the World Cup, I can feel that it will be best for the team that there is a new face and new ideas who can write a new big chapter for this fantastic team, and therefore it is best that someone else leads the team forward.
"It's not about me. It's about what's best for the team."
OneLove, which promotes 'inclusion and sends a message against discrimination of any kind', has grown in significance in the build-up to the tournament in Qatar, a country in which homosexuality is illegal.
England, along with other European nations like Germany, Denmark and the Netherlands, formed an alliance with all stating their captains would wear the armband in a show of solidarity and support with the LGBTQ community.
However, FIFA attempted to wrestle back control of the narrative on Saturday by launching their own collection of armbands across a range of social issues with a different subject for each stage, including 'Save the Planet' and 'Bring the Moves'.
It also emerged they were also considering ordering referees to issue yellow cards to captains wearing the OneLove armbands as soon as matches kicked off.
It has left the Football Association (FA) in an uncomfortable position just hours before Gareth Southgate's men open their campaign at the Khalifa International Stadium in Doha.
FA chief executive Mark Bullingham told Radio 4's Today programme: "We've had meetings with FIFA this morning and there are discussions that are carrying on.
"We're very keen to wear the armband, we want to do it, but we would need to consider the implications.
"Normally in these situations there'd be a fine that would get paid and we've always said we are very happy to do that – well, happy might be the wrong word but we'd be prepared to pay the fine because we think it's important to show our support for inclusion.
"If the sporting sanction threat is real, though, we'd need to look at that, take a step back and work our if there's another way to show our values."
The world's top-ranked side were without Kevin De Bruyne and Thibaut Courtois for their trip to Copenhagen, while Eden Hazard remained on the bench for the whole game.
Despite the notable absentees, Belgium picked up an 11th win in a row in their first game back after a 10-month hiatus thanks to goals from Jason Denayer and Dries Mertens.
Martinez was pleased to see some of his fringe players step up as the Red Devils made a winning start to their Group A2 campaign.
"After 10 months without playing together, we needed to find some rhythm," he told RTL Sport.
"Despite everything, we managed to remain threatening, scored a good goal and maintained a good attitude throughout. I am delighted with this performance.
"We miss players like De Bruyne, Hazard and Courtois, and yet the team has shown that they always know how to win. It's incredible!
"There will probably be three changes in the next game because some players can't play longer than 90 minutes.
"Thomas Meunier will probably play. But we won't make too many changes because we have to create some rhythm."
Denayer, who scored his first international goal on his 14th appearance, echoed the views of his head coach after a hard-fought victory at the Parken Stadium.
"It was difficult at first, but we regained our rhythm as the match progressed," he said. "We had players missing, and yet we have still won. It shows that the squad is very strong."
Denmark struggled to create many opportunities across the 90 minutes, with stand-in Belgium goalkeeper Simon Mignolet making just one save of note to keep out Christian Eriksen.
New head coach Kasper Hjulmand was happy with the overall display of his side but admitted the visitors' quality told in the end.
"I think we played a really good first half," he told Kanal 5. "When Belgium go in front, they are hard to deal with. They stood well and were good at defending.
"Otherwise, I think our first half was perfectly okay. We played with a great deal of intensity and got in good positions. We stopped them creating the same number of chances as usual.
"We handled the first half really well, and there were also 10-12 minutes of the second half, where I think we did what we were supposed to.
"Then we ran out of intensity and energy. And then you could see the quality of the Belgian team."
