WASHINGTON — Senate Republicans on Thursday thwarted the confirmation of two of President Obama’s nominees, one to a powerful appeals court and another to a home-lending oversight post, setting up a confrontation with Democrats that could escalate into a larger fight over limiting the filibuster and restrict how far the minority party can go to block a president’s agenda.

In a series of swift back-to-back votes, Republicans first blocked the nomination of Representative Melvin Watt, Democrat of North Carolina, to become the director of the Federal Housing Finance Agency — a rare affront to a sitting member of Congress who has an extensive record of public service.

Next Republicans, who have accused the president of trying to tip the court’s ideological balance in Democrats’ favor, quickly dispensed with the nomination of Patricia Ann Millett to the United States Court of Appeals for the District of Columbia Circuit. A former government lawyer whose husband serves in the military, she has worked in both Republican and Democratic administrations. The White House chose her as a test of how far Republicans would go to derail a qualified nominee.

Vice President Joseph R. Biden Jr., a longtime member of the Senate and a fierce protector of its arcane institutions, said he believed the rejection of the two nominees was grounds to re-examine the filibuster rules, which some senior Senate Democrats have advocated.