House Republicans on Friday thrust President Barack Obama's health law into the middle of two looming fiscal battles, a strategy that roils Congress's efforts to keep the government solvent this fall and avoid a partial government shutdown in less than two weeks.

The House on Friday passed a Republican bill to keep the government funded starting Oct. 1. That measure, however, also eliminates money for the health-care law. It was a triumph for a wing of the Republican Party that has campaigned for months to convince GOP leaders...