She has dismissed the suggestion of such a land deal as “absurd”.
What did the US president say?
Speaking to reporters on the White House lawn on Wednesday afternoon, Mr Trump took umbrage at Ms Frederiksen’s remarks.
“I thought that the prime minister’s statement that it was absurd, that it was an absurd idea was nasty,” he said.
“I thought it was an inappropriate statement. All she had to do is say no, we wouldn’t be interested.”
He continued to make digs at Denmark online.
For the record, Denmark is only at 1.35% of GDP for NATO spending. They are a wealthy country and should be at 2%. We protect Europe and yet, only 8 of the 28 NATO countries are at the 2% mark. The United States is at a much, much higher level than that….
What did the Danish PM say?
Earlier on Wednesday, Ms Frederiksen reiterated that Greenland could not be bought.
She told reporters the idea of selling the resource-rich Arctic island had “clearly been rejected” by its leader, Kim Kielsen, “a position I share of course”.
“This does not change the character of our good relations and we will continue our dialogue on how we can deal with challenges we are facing,” she said, adding that the invitation to Mr Trump “remains open”.
She has said Mr Trump’s no-show was a matter of regret because “our preparations were well under way”.
How did we get here?
While praising Denmark as a “very special country”, Mr Trump said in a tweet on Tuesday that his planned visit would no longer go ahead because Ms Frederiksen had “no interest in discussing the purchase of Greenland”.
Mr Trump had earlier confirmed reports that he was interested in buying Greenland. When asked on Sunday if he would consider trading a US territory for the island, he replied: “Well, a lot of things could be done.”
“Essentially it’s a large real estate deal,” he said.
On Monday, the US president tweeted a jokey image showing a tall golden skyscraper among the homes of a small village on the island.
How have Danes reacted?
The cancellation was described as a “farce” by the leader of the populist Danish People’s Party, Kristian Thulesen Dahl.
“What is this man thinking of though? And with grounds that are worthy of an April Fools’ joke,” he tweeted.
Danish Conservative MP Rasmus Jarlov accused Mr Trump of lacking respect for his country.
Former foreign minister Kristian Jensen said Mr Trump’s move had resulted in “total chaos”.
A spokeswoman for the leftist Red-Green Alliance, Pernille Skipper, said: “Trump lives on another planet.”
Pia Kjaersgaard, the populist former speaker of the Danish parliament, said it showed “rude behaviour to the Danish people and the Queen, who invited him.”
The US has long seen the island, which sits along a direct route from Europe to North America, as being strategically important. It established the Thule air force and radar base there at the start of the Cold War, which now covers space surveillance and forms the northernmost part of the US ballistic missile early warning system.
Meanwhile, new Arctic sea routes are opening up as climate change is blamed for the accelerating thaw of ice in the region.
China has recently been taking an interest in the area, too.