WASHINGTON — President Obama’s latest choice to fill one of the vacancies on a powerful appeals court went down in a filibuster on Tuesday as Senate Republicans blocked another White House nominee — the third in two weeks — and deepened a growing conflict with Democrats over presidential appointments.

By a vote of 56 to 41, the nomination of Cornelia T. L. Pillard, a Georgetown law professor, fell short of clearing the necessary 60-vote threshold.

Ms. Pillard’s liberal record on issues like abortion has troubled many conservatives, who are concerned about her attaining a position on a court widely considered second in stature to only the Supreme Court. But Senate Republicans have tried to frame their opposition to her not in terms of her views on social issues, instead focusing on the caseload at the court, the United States Court of Appeals for the District of Columbia Circuit, which is lower than that of other federal appeals courts.

The disagreements carried over onto the Senate floor on Tuesday, as Democrats accused Republicans of blocking a perfectly qualified woman for political purposes, while Republicans said Democrats were desperately looking for a wedge issue.