But Lee said he is hearing from colleagues that their constituents support blocking enactment of the law by denying funding to implement it, but some senators, including party leaders, have not responded.
"When you know something is necessary in order to protect the people from something that could be very harmful for them, you've got to stand for it, even when it's difficult, when the party establishment in your own party is against it," Lee said. "There is a certain sense where this is not a simple clash between Democrats and Republicans or even liberals versus conservatives. It's the ruling class in Washington versus the American people."
Lee cringes at those, including those in his own party, who say he wants to risk shutting down government by refusing to pass a budget that includes funds for Obamacare.
If the budget bill doesn't pass, government would shut down and it is highly unlikely that Senate Democrats would pass a bill or President Barack Obama would sign a bill that denies funding to the president's signature health care program.
No means no • Lee said it is "a dang lie" that he wants to shut down the federal government and said it shouldn't have to come to that. But both he and Cruz acknowledged that if either was in a position to cast the deciding vote between funding Obamacare and shutting down the federal government, they would vote for the latter.
"Under no circumstances will I vote for a continuing resolution that funds Obamacare and the reason is simple: It's not working, it's hurting the American people," Cruz said.
This fight is the latest example of attempts by Cruz, Lee and others branded as tea-party conservatives, to challenge the party establishment and reshape the GOP, a move political experts say comes with some risk.
Cruz was in Utah Friday to help raise campaign cash for Lee, who helped with Cruz's Senate bid in 2012 ideological brethren supporting one another and, in Cruz's case, potentially building a national network for a possible presidential run.
Cruz wouldn't say what his plans are for 2016, only that he is focused on the Senate and fighting Obamacare.
'Off-center' • "So far, their bark is worse than their bite," said Julian Zelizer, a professor of history and public affairs at Princeton University.
He said Cruz and company have been adept at getting media attention, but they have failed to dramatically change the party's direction.
"Ultimately, it's unclear if they could deliver nationally and it's unclear, if they have a presidential candidate like Cruz, that he is going to win," said Zelizer. "They're a big outlier. A lot of their positions are off-center with where the general populace is and even where a lot of Republicans are. How do you make them the voice of the Republican Party without marginalizing the party?"
Thad Hall, a professor of political science at the University of Utah, said the coming months will be a test for Cruz, Lee and their cohort, as the debt-ceiling vote nears, as well as a likely vote on a resolution to fund the government that will put the anti-Obamacare effort to the test.
"These are the people who are going to want to shut down the government to make sure we don't have Obamacare and they're not going to want to raise the debt ceiling, so they're going to put the party in an extreme position," said Hall.
But Cruz said it is essential that Republicans stand for conservative issues.
"When Republicans paint in pale pastels, when Republicans run as Democrat-lite, we lose," he said. "When Republicans draw a clear distinction, when we embrace economic growth, opportunity and the constitutional principles that form the foundation of this country, we win."
Republican leaders are increasingly alienating tea-party activists, whose disapproval of GOP leaders is now at 71 percent, with approval at just 27 percent, according to a Pew Research Center Survey released earlier this week.
Another Pew survey this summer found that rank-and-file Republicans want to see their party move to the right. Fifty-four percent said the GOP needs to be more conservative, while 40 percent said it should become more moderate.
Lee, meantime, is facing some uncertainty in his home state. His approval rating has been lackluster typically lower than other members of the congressional delegation and well below other statewide officials.
Through the end of June, he had $72,860 in his campaign account and owed $58,177 in consulting fees.
Former Utah Republican Party Chairman Thomas Wright has said he is considering challenging Lee when he's up for election in 2016.