WASHINGTON—U.S. Circuit Judge Merrick Garland’s nomination to the Supreme Court expired at noon Tuesday, clearing the way for President-elect Donald Trump to fill a vacancy Senate Republicans held open for months with an appointee championed by conservatives.

Judge Garland’s nomination languished for nearly 10 months until it expired with the formal adjournment of the 114th Congress. Republicans said their inaction on the nomination was a way to permit voters to weigh in on the Supreme Court’s direction by electing the next...