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.

The 51 to 48 vote to kill the amendment was largely along party lines, although three Democrats — Robert P. Casey Jr. (Pa.), Joe Manchin (W.Va.) and Ben Nelson (Neb.) — broke ranks to support it, and Republican Olympia Snowe (Maine) opposed it.

All the Republican candidates have made the birth control issue a key part of their stump speeches in recent weeks, banking on conservative anger over what they contend is the rule’s narrow religious exception.

Only churches are fully exempted, although under an accommodation recently announced by the Obama administration, religiously affiliated organizations such as Catholic charities, schools, universities or hospitals can refuse to provide contraceptive coverage through their insurance plans for employees. In such cases, employers’ insurance companies must offer coverage to female employees directly, without charging additional premiums.

For the most part, the Republican candidates have focused on President Obama, casting him as hostile to religious freedom. But at a rally in Georgia Thursday, former senator Rick Santorum (Pa.) accused former Massachusetts governor Mitt Romney of being insufficiently committed to the conservative position.

Santorum contended that Romney’s instinctive reaction to a TV interviewer’s query the day before had been to oppose the Blunt amendment.

“When Governor Romney was asked that question, his knee-jerk reaction was, ‘Oh, I can’t be for it,’ ” Santorum said. “Well, then after his consultants talk to him, he said, ‘Well, I didn’t understand the question.’ Well, maybe he did, maybe he didn’t. If I was asked that question, my gut reaction . . . would be, ‘You stand for the First Amendment; you stand for freedom of religion.’ ”

During that television interview in Ohio on Wednesday, Romney was asked for his view of what the interviewer described as a “Blunt-Rubio bill ... banning or allowing employers to ban providing female contraception.”

Romney responded that he was opposed.

“I’m not for the bill, but look, the idea of presidential candidates getting into questions about contraception within a relationship between a man and a woman, husband and wife, I’m not going there,” Romney answered.

The response immediately led to headlines that Romney had abandoned Republican leaders, including Blunt, who serves as Romney’s congressional liaison, on a key social issue.

Romney’s campaign quickly said he had been confused and thought he was being asked to respond to a local proposal in Ohio. “I didn’t understand his question,” Romney said on a Boston radio show later Wednesday. “Of course, I support the Blunt amendment.” And in a statement Thursday, Romney praised senators who backed the measure.

“The president of the United States must protect and defend the Constitution, not ignore it,” he said. “This is yet another example of what is wrong with Obamacare, and why I am committed to its repeal.”

Blunt accepted that explanation, telling reporters Thursday that the interview question, “was about as confusing and disjointed as you could be, and he [Romney] quickly clarified that.”

Still, the damage had been done. Romney’s Republican rivals and the Obama campaign immediately accused him of reversing positions in a matter of hours, reinforcing his critics’ depiction of him as wishy-washy on core social issues.

The fumble could hurt Romney among socially conservative voters in key Super Tuesday states, including Georgia, Tennessee and Ohio.

In general, both parties appear to see the controversy as a winning issue. To Republicans, it offers a way to excite the base and paint the administration as overly intrusive and out of touch with Americans’ religious sensitivities. To Democrats, it is an opportunity to portray Republicans as willing to trample women’s rights.

On Thursday, Vice President Biden said the administration had initially flubbed the issue before regaining its footing by announcing the accommodation for religiously affiliated employers.

“It got screwed up in the first iteration,” said Biden at Iowa State University.

Still, Democrats feel they now hold the political high ground, and they said the vote on the Blunt amendment strengthened their position.

In speech upon speech on the Senate floor Thursday, Democrats contended that the moral exemption in the amendment was so “extreme” that employers might use it to refuse to cover immunizations, prenatal care for children conceived out of wedlock or HIV screenings for gay employees.

“It would simply give every boss in America the right to make the health-care decisions for their workers and their families,” said Sen. Patty Murray (D-Wash.). “It is a radical assault.”

In a news conference after the vote, several Democrats cast the measure as framing a key issue in the November elections, suggesting that Republican control in Washington would mean a reversal of Thursday’s outcome.

“The closeness of this vote shows how high the stakes are for women. A Republican-led Senate might pass this bill. A Republican president, like Mitt Romney, would definitely sign it,” Sen. Charles E. Schumer (D-N.Y.) said.

“If Republicans keep this up, they’re going to drive away independent voters, women and men, just as they are driving moderates out of their caucus,” he said, an allusion to Snowe’s decision this week to retire from the Senate. Snowe has decried partisanship on both sides of the aisle.

Sen. Susan Collins, Snowe’s fellow Maine Republican, also expressed discomfort over the issue. She said she felt compelled to support the Blunt amendment on behalf of religiously based groups that self-insure and, therefore, cannot take advantage of the administration’s proposed accommodation to the rule. But Collins added that she was “conflicted” over the vote and called for the Senate to move quickly to other issues.

Staff writers Phil Rucker, in North Dakota, and Nia-Malika Henderson, in Georgia, contributed to this report.