The Senate Thursday rejected an effort to vastly expand conscience exemptions to the Obama administration’s new birth control coverage rule, even as Republican presidential contenders continued to tussle over the issue.

The measure, an amendment proposed by Sen. Roy Blunt (R-Mo.) to a highway funding bill, would have allowed not only religious groups but any employer with moral objections to opt out of the coverage requirement. And it would have allowed such employers to do so in the case of not only contraception but any health service required by the 2010 health-care law.