WASHINGTON—Senate Republicans voted to end the filibuster of Supreme Court nominations Thursday, setting the stage for the rapid elevation of Judge Neil Gorsuch to the high court and removing a pillar of the minority party’s power to exert influence in the chamber.

Judge Gorsuch’s confirmation by the Senate, expected Friday, would return the Supreme Court to full strength for the first time in 14 months, since the death of Justice Antonin Scalia in February of last year. It means Judge Gorsuch could be a key vote on coming...