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Along those lines, he joined former Sen. Jim DeMint of South Carolina, who now runs the conservative Heritage Foundation, at anti-Obamacare town-hall meetings over the summer that were meant to ramp up pressure on lawmakers of all stripes to join the fight — or else.

Mr. Cruz followed up last month with a 21-hour filibuster against a Senate measure that included Obamacare funding, telling fellow Republicans that anyone who allowed the resolution to move forward would be voting to “fully fund Obamacare.”

The effort failed to move the needle in the Senate, but it did inject some energy into House Republican efforts to put the brakes on the health care law.

The House has stonewalled a Senate bill that would fund all government operations and Obamacare through Nov. 15, which Democrats say would give both parties a chance to work out their differences.

Over the past week, the House has pushed individual bills to fund specific government programs, including Veterans Affairs and museums, while demanding that Democrats accept a one-year delay of Obamacare’s individual mandate.

As much as Mr. Cruz says the fight is not about him, his own messaging at times seems to contradict that.

Last week, he took to the floor of the Senate to call on the chamber to act on the House bills, a day after Senate Republican leaders made the same request.

Both initiatives failed and the Cruz effort fed into lingering Republican concerns that Mr. Cruz’s strategy is misguided. Mr. Reid pointed out that Rep. Devin Nunes, a California Republican, said House Republicans were waiting for Mr. Cruz to explain the next move now that “he’s the one that got us into this mess.”

Other Republicans are more reluctant to weigh in on the Cruz effect. Last week, Rep. Michael K. Simpson of Idaho laughed and suggested he was better off keeping his thoughts to himself before slipping into the House chamber for a vote.

Still, Mr. Cruz has his followers, most notably among House Republicans who are calling on Speaker John A. Boehner of Ohio to insist that Obamacare does not get off the ground.

“I think he has been a huge asset,” said Rep. John Fleming, Louisiana Republican. “He’s been a rallying force. I think he has been a unifying force and has been an energizing force. I think he has had a huge positive impact.”

Rep. H. Morgan Griffith, Virginia Republican, said Mr. Cruz’s influence has been good and bad.

“I think the positive is that he has brought a lot of attention to the issue,” Mr. Griffth said. “The negative is that he stepped on some toes in the process. But as the old saying goes, in order to make an omelet, you have to break some eggs.”

Democrats, meanwhile, are happy to use Mr. Cruz as an all-purpose Republican boogeyman, hoping that it will bolster their argument in the spending debate and help the party pick up seats in the 2014 elections.

“I think Ted Cruz represents an ideology that is so far out of the mainstream that, quite frankly, responsible Republicans here should be ignoring him just like responsible Republicans in the Senate are ignoring him,” said Rep. James P. McGovern, Massachusetts Democrat. “He doesn’t represent serious legislating. He is just a guy that likes to get up and say provocative things and rail against government. [Former House Speaker] Sam Rayburn used to say that any jackass can kick down a barn, but it takes a real carpenter to build one.”