Trump, who takes office on January 20, also floated the idea of turning Canada into a US state, said he would demand far higher defence spending from NATO allies and promised to change the name of the Gulf of Mexico to the Gulf of America.
Still two weeks away from taking office, Trump has begun outlining an aggressive foreign policy with little regard to diplomatic considerations or the concerns of US allies.
Asked at a press conference on Tuesday at his Florida resort whether he could assure the world he would not use military or economic coercion as he tries to gain control of the Panama Canal and Greenland, Trump said: "No, I can't assure you on either of those two. But I can say this, we need them for economic security."
Trump criticised American spending on Canadian goods and military support for Canada, saying the US derives no benefits from doing so, and called the border between the two countries an "artificially drawn line".
He suggested he would impose tariffs on Denmark if it resists his offer to purchase Greenland, which he said is vital to US national security. Shortly before Trump's comments, his son Don Jr. arrived in Greenland for a private visit.
Denmark has said Greenland, a self-governing part of its kingdom, is not for sale.
"I don't think it's a good way forward to fight each other with financial means when we are close allies and partners," Denmark's Prime Minister Mette Frederiksen said late on Tuesday in response to Trump's comments.
Canada's minister of foreign affairs, Melanie Joly, said on X, "President-elect Trump's comments show a complete lack of understanding of what makes Canada a strong country. Our economy is strong. Our people are strong. We will never back down in the face of threats."
Panama's top diplomat also pushed back on the incoming US leader's threat to retake the key global waterway, which the US had built and owned before handing over control to the Central American nation in 1999.
"The only hands that control the canal are Panamanian and that's how it will continue to be," Foreign Minister Javier Martinez-Acha told reporters on Tuesday.
Ambassador Daniel Fried, a retired US diplomat now with the Atlantic Council think tank, said Trump's comments painted a picture of national power as territorial expansion and compared him to a "19th century imperialist".
Seizing Greenland, Fried said, "would destroy NATO, because it would make us no different than Vladimir Putin," Russia's president.
Trump's promise to rename the Gulf of Mexico echoed his previous vow to revert the name of Denali, the highest mountain peak in North America, to Mount McKinley.
Former President Barack Obama changed the name of the Alaskan mountain in deference to Native Americans.
Typically, the US Board of Geographic Names sets geographic names, though presidents have also renamed geographic features via executive action.
The free-wheeling, hour-long press conference, Trump's second since his victory, echoed similar events during his presidential campaign.
He aired a series of familiar grievances about his criminal indictments, including attacking Justice Juan Merchan, the New York judge who is scheduled to sentence Trump on Friday for falsifying business records in connection with hush money payments to a porn star.
A New York appeals court denied his latest bid to halt the sentencing shortly after his press conference ended.
