WASHINGTON—Sen. Ted Cruz came up empty-handed Wednesday in his battle to gut the Affordable Care Act, saying that he wouldn't stand in the way of a Senate budget deal that includes none of the major changes to the health-care law that conservatives were seeking.

Mr. Cruz's high-profile fight against the law known as Obamacare kicked off last month with a headline-grabbing 21-hour speech, and on Wednesday, it appeared likely to end—at least for the moment—with a political defeat for the Texas Republican.

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