Trump said on Saturday on social media he was working with Louisiana Governor and special envoy to Greenland, Jeff Landry, to send a hospital boat to Greenland.
"President Trump's idea of sending an American hospital ship here to Greenland has been noted. But we have a public healthcare system where treatment is free for citizens. It is a deliberate choice," Nielsen said in a post on Facebook on Sunday.
Nielsen said Greenland remained open to dialogue and cooperation, also with the US.
"But talk to us instead of just making more or less random outbursts on social media," he said.
Greenland, Denmark and the US late last month launched diplomatic talks to resolve the crisis between the parties, following months of tension within the NATO defence alliance over Trump's threats against the Arctic territory.
Trump's post on the ship came hours after Denmark's Joint Arctic Command said it had evacuated a crew member who required urgent medical treatment from a US submarine in Greenlandic waters, seven nautical miles outside of Greenland's capital, Nuuk. It was unclear if the post had any connection to the evacuation.
The crew member was retrieved by a Danish Seahawk helicopter deployed on an inspection ship.
Denmark's Prime Minister Mette Frederiksen defended Denmark's health care system on Sunday, writing on Facebook that she was "happy to live in a country where there is free and equal access to health for all".
