The acting attorney general on Monday ordered Justice Department lawyers not to defend President Donald Trump’s executive order suspending immigration from seven majority Muslim countries—a remarkable move that likely will lead to a showdown between the Republican White House and the holdover appointee from the Obama administration.

The decision by Sally Yates, who became the acting head of the Justice Department when her boss, Loretta Lynch, stepped down as attorney general this month, is likely to provoke an intense albeit...