"That won't happen," Danish Defence Minister Trouls Lund Poulsen told public broadcaster DR.
"The direction that Greenland wants to take will be decided by Greenlanders," Lund Poulsen said.
In a partisan speech to Congress on Tuesday, Trump offered only passing lines on world affairs, focusing on his domestic goals like rounding up undocumented immigrants and slashing government spending.
But he underlined his expansionist vision of the United States, as he repeated his aspirations to take Greenland and claimed an initial victory on retaking control of the Panama Canal.
One week before general elections in Greenland, an autonomous territory of Denmark with an independence movement, Trump said he had a message for the "incredible people" of the sparsely populated but mineral-rich and strategically-placed island.
"We strongly support your right to determine your own future, and if you choose, we welcome you into the United States of America," Trump said.
But he made clear he would not give up if persuasion fails, saying: "One way or the other we're going to get it."
"We will keep you safe, we will make you rich, and together, we will take Greenland to heights like you have never thought possible before."
China and Russia have been stepping up activity in the Arctic as climate change opens further sea routes.
- 'Taking it back' -
Danish Foreign Minister Lars Lokke Rasmussen called for cool heads to prevail.
"I think everyone, including us, should be cautious about having all kinds of opinions about the future" of Greenland, he told Danish television TV2.
US threats to take Greenland would once have been thinkable, with Denmark a treaty ally of the United States under NATO.
But Trump has made clear he has little patience for European allies, which he again denounced for not spending more on their militaries, with Trump instead seeing a return to an era of big powers taking what they want.
He has similarly vowed to take back the Panama Canal, the crucial link between the Atlantic and Pacific oceans that the United States handed to Panama at the end of 1999.
Trump declared triumph after Hong Kong firm CK Hutchison decided to sell its Panama ports to a US-led consortium.
Trump and Secretary of State Marco Rubio had complained that rival China had gained too much influence over the canal and could shut it down in a conflict with the United States.
"To further enhance our national security, my administration will be reclaiming the Panama Canal, and we've already started doing it," he said, as he mentioned the port deal.
"We didn't give it to China. Gave it to Panama -- and we're taking it back," he said.
Trump had earlier not ruled out military force to seize either the Panama Canal or Greenland.
Trump has paradoxically sought to cast himself as a peacemaker. He has vowed to end the war in Ukraine and has rattled allies by suspending aid to the country, which Russia invaded three years ago.
Trump and Vice President JD Vance berated Ukrainian President Volodymyr Zelensky when he visited the White House on Friday, calling him ungrateful.
Addressing Congress, Trump read a message from Zelensky in which the Ukrainian leader sought to repair the damage and voiced a willingness to a sign a deal in which the United States would take much of Ukraine's mineral wealth.
"It's time to end this senseless war. If you want to end wars, you have to talk to both sides," Trump said.
- Overseas aid -
The Trump administration at the same time has canceled more than 90 percent of US development assistance, traditionally a key source of US non-military influence.
Trump has described aid as not in the US interest, with his advisor Elon Musk, the world's richest person, pushing massive cuts to make way for tax cuts.
Trump rattled off a list of previous aid grants and reserved mockery for a project in the landlocked African country of Lesotho, which has one of the world's highest rates of HIV/AIDS.
"Eight million dollars to promote LGBTQI+ in the African nation of Lesotho," Trump said, struggling with the pronunciation.
"Which nobody has ever heard of," Trump added to laughter from Republican lawmakers.
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