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Speaking on the chamber floor, Sen. Tom Coburn, Oklahoma Republican, said motives of Mr. Cruz and his allies are “pure,” but “their tactics are tremendously erroneous.”

Mr. McCain said Republicans should push to replace and repair Obamacare, but that he disagrees with Mr. Cruz’s approach. A spokesman for Orrin G. Hatch, Utah Republican, said the veteran lawmaker supports Mr. McConnell’s strategy.

Sen. Charles E. Schumer, New York Democrat, said Mr. Cruz’s effort backfired.

Sen. Cruz has actually advanced our cause,” Mr. Schumer said. “He has alienated some of his colleagues, he has united Democrats and he has shown the American people that he is willing to hold them and their well-being hostage unless he gets his way.”

Jacqueline Klimas contributed to this report.