BRUSSELS—President Donald Trump’s decision to impose tariffs on steel and aluminum from the European Union marks a new low in deteriorating trans-Atlantic relations and promises to complicate life for EU leaders grappling with growing problems inside the bloc.

Coming just weeks after Mr. Trump’s withdrawal from the Iran nuclear deal and a day short of a year after his exit from the Paris climate accord, it offers another sign that the U.S. administration has broken with decades of American policy by playing down European...