Greenland has urged Donald Trump to give up his “fantasies about annexation” after the US president, fresh from his military operation in Venezuela, again threatened to take over the Arctic territory.

In a bracingly direct statement, the Greenlandic prime minister, Jens-Frederik Nielsen, accused the US of “completely and utterly unacceptable” rhetoric, declaring: “Enough is enough.”

On Sunday, Trump said the US needed Greenland “very badly” – renewing fears of a US invasion of the largely autonomous island, which is a former Danish colony and remains part of the Danish kingdom. Greenland’s foreign and security policy continues to be controlled by Copenhagen.

“Threats, pressure and talk of annexation have no place between friends,” said Nielsen in a social media post on Monday. “That is not how you speak to a people who have shown responsibility, stability and loyalty time and again. Enough is enough. No more pressure. No more innuendo. No more fantasies about annexation.”

Greenland, he said, was “open to dialogue” but it had to come through the appropriate channels and in line with international law, “not random and disrespectful posts on social media”.

He added: “Greenland is our home and our territory. And that is how it will remain.”

He was backed by the EU, which on Monday said it would not stop defending the principle of territorial integrity, particularly when it came to a member of the 27-member bloc.

“The EU will continue to uphold the principles of national sovereignty, territorial integrity and the inviolability of borders,” EU foreign policy spokesperson Anitta Hipper told reporters.

“These are universal principles, and we will not stop defending them, all the more so if the territorial integrity of a member state of the European Union is questioned.”

The comments come after Denmark’s prime minister, Mette Frederiksen, on Sunday said it made “absolutely no sense to talk about the US needing to take over Greenland”. She added that the US had “no right to annex any of the three countries in the Danish kingdom”.

After refusing to rule out military intervention to gain control of Greenland last year, Trump has been relatively quiet on the subject in recent months.

But the US bombardment of Venezuela to capture its president, Nicolás Maduro, and Trump’s comments over the weekend have renewed fears that he may act on his threats.

Speaking on Air Force One, asked whether he expected to take action in Greenland, he refused to answer, saying he would revisit the subject “in 20 days” before going on to mock Danish defence efforts.

“Right now, Greenland is full of Chinese and Russian ships everywhere. We need Greenland for national security reasons. Denmark will not be able to handle the task.”

Last month, Danish intelligence services accused the US of using its economic power to “assert its will” and threatening military force against its allies. It comes against the backdrop of increasing tensions in the Arctic as three global superpowers – the US, China and Russia – fight for dominance over minerals and other geopolitical assets in the region as the ice melts.

Nordic neighbours Sweden, Norway and Finland have all expressed their support for Denmark. The Swedish prime minister, Ulf Kristersson, said: “Only Denmark and Greenland have the right to decide on issues concerning Denmark and Greenland. Sweden fully supports our neighbouring country.”