The provision is basically a further application of the Hyde Amendment, which generally prohibits taxpayer funds from being used for abortions. Democrats say they do not like what they see as an expansion of the Hyde Amendment to include “fees” and a precedent that could expand abortion restrictions.

Democrats, many of whom voted for the bill originally in committee, further insisted that Republicans had sneaked the abortion language into the bill after that vote, a charge Republicans denied. They declined to remove the language.

The procedural bill fell five votes short of the 60 needed to go forward, and Senator Mitch McConnell of Kentucky, the majority leader, said Tuesday that he would bring the bill to the floor again this week — as early as Wednesday — and try to enlist the additional Democrats needed to move forward. He also held strong to his position that he would not turn to the matter of confirming Ms. Lynch, Mr. Obama’s nominee to replace Attorney General Eric H. Holder Jr., until the sex-trafficking bill got off the floor.

“We’re going to stay on this bill,” he said Tuesday.

Four Democrats voted yes on the measure with their Republican colleagues. They were Senator Heidi Heitkamp of North Dakota, who has long championed measures for victims of sex trafficking, and Senators Bob Casey of Pennsylvania, Joe Donnelly of Indiana and Joe Manchin III of West Virginia, who are opposed to abortion.