The new Senate rule clears the way for eight appeals court nominees who have already had confirmation hearings to win approval with simple majority votes, including three on the powerful Court of Appeals for the District of Columbia Circuit, which reviews federal policies and regulations. But it left unchanged the Senate’s “blue slip” custom, which allows senators to block nominees to judgeships associated with their states.

“It is hard to overstate the change’s importance for the D.C. Circuit, which has a disproportionate impact on the world, but it won’t have overwhelming impact elsewhere,” Kathryn Ruemmler, the White House counsel, said in an interview. “The blue slip rule for judges has been more problematic than the filibuster, in part because it is a silent, unaccountable veto.”

Twelve more appeals court seats are either vacant or will be by the end of 2014. All but one are in states with at least one Republican senator. As a result, Mr. Obama still lacks unrestricted power to swiftly appoint a flurry of more clearly left-of-center judges than he has done to date, despite the fears of conservatives and the hopes of liberals, specialists said.