The outcome of the Michigan primary Tuesday failed to clarify Rick Santorum’s standing in the Republican presidential race and left open a central question:

Did his strident rhetoric on social issues light a fire among GOP voters that could boost his chances of winning the nomination, or did it expose a vulnerability that will ultimately lead Republicans to decide that Mitt Romney is the safer bet to take on President Obama in November?

Romney scored a modest ­victory over Santorum in Michigan and easily won Arizona, which Santorum had essentially ceded.

Even before the polls closed, Santorum said his ability to run close in Michigan, where Romney was born and his father served as governor, was a sign of his opponent’s fatal flaws and an indication of the enduring strength of his own candidacy.

“A month ago, they didn’t know who we are. But they do now,” Santorum told supporters in Grand Rapids, Mich., on Tuesday night. “We came into the back yard of one of my opponents, in a race that everyone said, ‘Well, just ignore, you really have no chance here.’ And the people of Michigan looked into the hearts of the candidates, and all I have to say is, I love you back.”

Santorum focused almost ­exclusively in his election-night remarks on promises to cut government spending, repeal Democratic health-care reform and revive the manufacturing sector. And for the first time, he stressed the role in his life of working women, including his mother, whom he said made more money than his father, a rarity in her generation.

In the days before the election, Santorum faced mounting scrutiny for claiming that Obama believes in a “phony theology,” for calling the president a “snob” because he urges children to go to college, and for saying John F. Kennedy’s famous 1960 address on the separation of church and state made him “almost throw up.”

None of the ideas were new for Santorum, but for the first time he got the attention devoted to a serious contender for the nomination. The results show that some Republicans may be starting to conclude that a Santorum candidacy is too risky for a party desperate to beat Obama among independents in November.

Exit polls indicated that more than half of Michigan voters thought Romney was the most likely candidate to defeat Obama; only a quarter said the same of Santorum.

“Republicans are listening for two things: they’re listening for ‘Do I agree with this?,’ and they’re listening for ‘Can this person win?’ ” said Michael T. Heaney, a professor of political science at the University of Michigan. “If they start to hear things that make the candidate sound implausible, that starts to hurt the candidate.”

Santorum has complained that the media have focused excessively on his comments about religion and social issues and ignored his economic message, including a plan to eliminate taxes for manufacturers that he credited with fueling his rise among blue-collar workers in Michigan.

But he has made those issues an integral part of his argument to voters. He has repeatedly argued that a conservative who sounds like he does is exactly what Republicans should be looking for as they seek someone who can beat Obama.

“Shock the establishment,” he told voters Monday in Kalamazoo, Mich. “They’re all worried, ‘Oh, this guy is too conservative.’ . . . We need someone who is going to, as Reagan did, remind us who we are. Spur the American public to do things. And believe big thoughts themselves.”

But Santorum signaled rare regret Tuesday for his heated rhetoric, telling conservative commentator Laura Ingraham that he wished he could take back his comment that he had wanted to vomit when he read Kennedy’s speech.

That remark may have especially hurt Santorum with Catholic voters, more of whom backed Romney than Santorum, according to exit-polling data, even though Santorum, like Kennedy, is Catholic.

Romney has reacted gingerly to Santorum’s more strident statements. His campaign in Michigan focused largely on painting Santorum as a Washington insider and a congressional big spender.

Only on Tuesday did Romney take direct aim at Santorum’s tone, accusing him of using “incendiary,” “outrageous” and ­“accusatory” comments to woo the party’s conservative base.

Romney’s caution may have come in part because the 10 states that vote next, on Super Tuesday, include Ohio, Oklahoma, Tennessee and Georgia — potentially friendly territory for Santorum’s appeal to social conservatives and his pugnacious attitude.

Santorum spent time Tuesday in Ohio and will travel to Tennessee on Wednesday. An independent super PAC working on his behalf is already airing television ads in Ohio and will be expanding to other Super Tuesday states in the coming days, said spokesman Stuart Roy.

“He’s been the front-runner, and you get an additional spotlight for better or for worse,” said Roy, a spokesman for a pro-Santorum super PAC that invested heavily in trying to convince Michigan voters that Santorum was a more conservative option than Romney. “You have to show you can take a punch and keep fighting. The best way to do that is to keep winning.”

Staff writers Nia Malika-Henderson and Philip Rucker in Michigan contributed to this report.