The association had expected well over a thousand people to take part, and Danes braved subzero temperatures en masse to support their veterans and the 44 Danes who died in Afghanistan.
Police told AFP they estimated the number of demonstrators were "at least 10,000", while organisers put the turnout at between 8,000 and 10,000.
Trump sparked outrage in Denmark and across Europe on January 22nd when he said European NATO troops "stayed a little back, a little off the front lines" during the 20-year conflict in Afghanistan. The Scandinavian country fought alongside US forces during the Gulf War as well as Afghanistan and Iraq.
Gathering in Copenhagen's Kastellet, or citadel, a brief ceremony was held at the monument to fallen soldiers before the procession began.
"The demonstration is called #NoWords because that really describes how we feel. We have no words," the vice president of the association, Soren Knudsen, told AFP.
"Obviously, we also want to tell Americans that what Trump said is an insult to us and the values that we defended together."
Some demonstrators waved red-and-white Danish flags and others were dressed in military uniform, as they marched quietly -- no slogans or chanting -- to the US embassy about two kilometres (1.2 miles) away.
Most were sombre, others were visibly emotional with tears streaming down their cheeks.
"We're very happy that so many people turned out," Knudsen told AFP outside the US embassy, pleased that veterans from the US and across Europe had also joined in.
"What Trump said was very insulting," Henning Andersen, who served as a Danish UN soldier in Cyprus, told AFP.
"I have friends who were down there. Some of them were wounded, and they carry the war with them even today," the 64-year-old said, four military medals pinned to his black veterans' jacket.
"He's saying things he doesn't know the full truth about."
One 58-year-old member of Denmark's home guard, who gave her name only as Orum, also expressed anger over Trump's remarks.
"How can he even say that? It's insulting," she said, clad in khaki fatigues and green beret.
44 flags
Protesters at the front of the march carried a large red banner reading "NoWords", while others carried hand-drawn signs. One said "Trump is so dumb", while another held by a child read "Say sorry, Trump".
In response to Trump's claim, 44 Danish flags, which carried the names of the 44 Danish soldiers killed in Afghanistan, were placed on Tuesday in planters outside the US embassy in Copenhagen.
The embassy removed the flags, before apologising and replacing them.
"We have nothing but the deepest respect for Danish veterans and the sacrifices Danish soldiers have made for our shared security. There was no ill intent behind the removal of the flags," the embassy said in a post on its Facebook page.
It noted that the planters were embassy property and not in the public domain, and the initial planting of the flags had not been coordinated with the embassy.
On Friday, the US ambassador placed 44 Danish flags in the flowerbeds.
On Saturday, 52 additional Danish flags were added, with names inscribed on them: 44 for the Danes who died in Afghanistan, and eight others for those killed in Iraq.
A minute of silence was also observed outside the embassy. One person laid down a wreath of red and white flowers.
Denmark has traditionally been an ardent US ally, and continues to call Washington its "closest ally" despite tensions over Trump's recent threats to seize Greenland.
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