ODENSE, Denmark — “I’m not going to enter a war of words with anybody, including the American president,” Denmark’s prime minister, Mette Frederiksen, told a Danish television channel on Wednesday.

Too late.

In Denmark, a small country with powerful neighbors, commitment to international alliances is bedrock policy, and polite, measured political debate is the norm. Ms. Frederiksen usually reflects that orthodoxy, but she has occasionally demonstrated a sharp tongue — by Danish standards — and skepticism about the United States; she made a splash early in her career by comparing American policy on women’s reproductive rights to Saudi Arabia’s.

So her tart dismissal this week of President Trump’s interest in buying Greenland as “an absurd discussion” was not surprising to Danes, and neither was her pivot to conciliatory words.