The European Union has been struggling to contend with fractious internal forces. Among them: the vote by Britain to leave the bloc, the organization’s failure to establish a unified response to the arrival of hundreds of thousands of asylum seekers and the debt crisis that has driven many Greeks into poverty. And then there are external pressures like Russia’s annexation of Crimea.

Before the election and since taking office, Mr. Trump has lauded the vote by Britain, known as Brexit, and said the country would thrive outside the European Union. He met with Nigel Farage, a populist leader of the campaign to leave the bloc, before seeing Prime Minister Theresa May. And at one point he went so far as to suggest that Mrs. May appoint Mr. Farage as Britain’s ambassador to the United States.

Mr. Trump has also praised President Vladimir V. Putin of Russia and indicated he would pursue friendlier relations with Moscow, even as Russia encourages chaos on the European Union’s eastern border.

Mr. Tusk’s letter does not reflect a new policy for the European Union, and member states of the 28-nation bloc are not required to act on Mr. Tusk’s advice when they meet on Friday. But many European leaders have made their differences with Mr. Trump known.

After the United States said it was temporarily blocking refugees from entering the country, Chancellor Angela Merkel of Germany felt compelled to point out to Mr. Trump the obligations of nations under the Geneva Conventions to protect refugees of war on humanitarian grounds. And President François Hollande of France said he had reminded Mr. Trump that “the ongoing fight to defend our democracy will be effective only if we sign up to respect to the founding principles and, in particular, the welcoming of refugees.”