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Mr. Cruz’s amendment was part of the debate over a massive spending bill to fund the government for the rest of this year. If it doesn’t pass, the government would shut down — leverage, some Republicans argued, that they could use to force Mr. Obama to cancel his health law.

But House Republican leaders, who control the lower chamber, declined to push that fight. Instead, they are attempting to eradicate Mr. Obama’s law in the long term as part of the normal budget process.

The fiscal-year blueprint released by Rep. Paul Ryan on Tuesday calls for the repeal of “Obamacare” in a bid to cut $4.6 trillion in spending and balance the nation’s books in 10 years.

Conservatives in the Senate said they wanted to see action now. In a floor speech that ranged from employment decisions at fast-food chains to his immigrant father’s economic struggles, Mr. Cruz said the health care law’s taxes, mandates and regulations will kill jobs and cause insurance premiums to soar.