In recent interviews here, Republican elected officials, voters and political strategists said the fact that Mr. Cruz and House Republicans lost their fight with the White House over Mr. Obama’s health care law was a side issue. What mattered, they said, was that Mr. Cruz, who had campaigned on shaking up the status quo in Washington, had fulfilled his promise. From local party leaders to county commissioners to Tea Party members, Mr. Cruz was praised for his courage.

“For a lot of us, this was refreshing,” said Mike Gibson, chairman of the Republican Party in Fort Bend County in suburban Houston. “We had a politician who said what he was going to do and then did it. Most Texas Republicans have been tired of our elected officials talking tough in Texas and then going to Washington and going along.”

Mr. Cruz, for one, said his Texas support has been uplifting. “The many supportive letters, e-mails, calls and social media comments we’ve received from Texans since Labor Day have been inspirational,” he said in a statement. “Hearing from constituents keeps me focused on the concerns of the people I work for and the issues I ran on. That’s what matters.”…

“I grit my teeth and bear it,” said a prominent Texas Republican, who spoke on the condition of anonymity because he said he wanted to avoid an intraparty fight. “I really hope he implodes sooner rather than later.”