President Obama moved to contain a growing political crisis Friday by altering a new birth control rule that had angered Catholics and ignited a debate across the political spectrum about the bounds of religious freedom.

The move represented an effort by the White House to walk a careful middle line that would retain the support of women’s groups and liberal lawmakers, who back the original rule, while easing tensions with Catholic critics, who reacted with cautious optimism and even praise to the shift.

The new policy did little to quell criticism from Republicans, who have been rallying opposition to the rule and using the issue on the campaign trail to paint Obama as hostile to religion.

The announcement caps a frenetic month in which the administration often sent mixed signals and appeared caught off guard by the political fallout as even allies questioned its handling of the issue.

Obama took the unusual step Friday of making a personal appearance in the White House briefing room to announce the decision. This amends a rule announced in August and set to take effect this summer that will require employers to provide contraceptive coverage with no out-of-pocket costs as part of their health plans for workers.

Churches have always been exempt from the mandate. However, Catholic and other religious leaders had complained that the rule would force church-affiliated institutions, such as schools, charities, hospitals and universities, to pay for health services that violated their beliefs.

Under the new arrangement — the details of which have yet to be finalized — women who work for such organizations would still be guaranteed contraceptive coverage. But they will obtain it directly from their insurance companies, which must provide the coverage without charging an additional premium.

“I’ve been confident from the start we could work out a sensible approach here,” Obama said. “Some folks in Washington may want to treat this as another political wedge issue, but it shouldn’t be. I never saw it that way.”

Obama’s involvement in what amounts to a technical regulatory matter underscored the political stakes for a president in the midst of a reelection campaign in which social issues have gained unexpected prominence.

As the White House struggled to resolve the contraception controversy Friday, across town, potential Republican challengers Mitt Romney and Rick Santorum were citing the birth control rule in stinging speeches to a gathering of conservative activists.

By contrast, several Catholic leaders, who had been pressing the administration for a broader religious exemption to the rule, seemed at least open to the plan. Cardinal-designate Timothy Michael Dolan of New York, president of the U.S. Conference of Catholic Bishops, who has led a national campaign on the issue, released a statement describing Obama’s decision as “a first step in the right direction.” He reserved judgement on the details.

Sister Carol Keehan, who heads the U.S. Catholic Health Association, went further, pronouncing herself “very pleased” in a statement. Shortly before his announcement, Obama called both Dolan and Keehan, as well as Planned Parenthood President Cecile Richards, aides said.

The decision raised new policy questions.

The administration has not decided how the arrangement will apply to religiously affiliated employers that self-insure, meaning that instead of purchasing a group policy on behalf of their workers, they collect premiums from workers directly, add their contributions to a pool and use the funds to pay for workers’ health-care costs as needed. In other words, cases in which the employer and insurer are effectively one and the same.

Self-insurance is popular among large employers, who often hire an insurance company to administer the fund. So a range of religiously affiliated employers are likely to fall into this category.

“We don’t really see how this new accommodation would satisfy our concerns,” said Michael Warsaw, president and chief executive of EWTN, a nonprofit nationwide network of Catholic television and radio stations that self-insures its roughly 350 employees. The organization has filed a lawsuit challenging the birth control rule.

An administration official, who requested anonymity to discuss what the White House is considering, said the White House will be looking at how to address such cases.

“This policy will be developed collaboratively,” the official said.

Other organizations expressed concern that employers with moral objections to contraception will not be eligible for the accommodation if they are for-profit enterprises.

Meanwhile, female staff members at churches, which are completely exempt from the rule, will not be able to obtain the additional coverage from their insurer.

Administration officials cited studies indicating that the impact of the accommodation on insurers would be cost-neutral. But industry representatives said they were troubled by the precedent the administration was setting in requiring them to provide a service without allowing them to adjust their premiums.

In addition to the escalating attacks from the GOP, the White House over the past week has faced intensifying pressure from friends on both sides of the issue.

Representatives of women’s groups and Democratic members of Congress, who were worried that the White House would go wobbly and weaken the rule, repeatedly called key Obama advisers to press their case.

Several leading Catholic Democrats and outside religious leaders who are otherwise supportive of the administration offered a competing perspective. They argued that the White House had needlessly put at risk the goodwill that Obama had built over years of courting religious leaders and voters.

By late last week, faith leaders close to the administration were pushing hard for a resolution.

“This thing was going south real fast,” said Joel C. Hunter, a Florida mega-church pastor who has advised the White House on faith issues and become a spiritual adviser to Obama. “I just can’t picture how they could fumble this kind of very important announcement.”

The fumble was months in the making and generated considerable internal strife within the White House.

The controversy first erupted in August, when Health and Human Services Secretary Kathleen Sebelius unveiled a proposed regulation fleshing out a provision of the 2010 health-care law. The law requires insurers to cover preventive care for women with no out-of-pocket charges, but it was left to Sebelius to draft the list of covered services.

Following the recommendations of an advisory panel of the National Academy of Sciences’s Institute of Medicine, Sebelius chose to include all forms of birth control approved by the Food and Drug Administration — including emergency contraception, such as Plan B and Ella, and sterilization.

She also proposed the exemption for churches, modeling it after similar provisions in several states.

Despite the subsequent avalanche of critical comments from Catholic and other religious groups, women’s advocates and Democratic lawmakers said that they assumed there was scant chance the administration would broaden the religious exemption.

However, Vice President Biden and then-White House Chief of Staff William M. Daley — Catholics with long-standing ties to church leadership — were sympathetic to the religious concerns, a White House official said.

Their influence became evident in early November, when Dolan, of the Conference of Catholic Bishops, raised the birth control issue with Obama in a personal meeting at the White House.

Dolan came away optimistic that Obama was considering a change to the policy, and he hinted at this in public comments soon after. His remarks stunned women’s groups and, according to the White House official, aides realized that the president had not been fully prepared ahead of his meeting with Dolan.

The administration’s efforts to respond were further complicated by the late 2011 departure of Melody Barnes, the top domestic policy adviser. Deliberations over the issue subsequently unfolded in haphazard hallway conversations.

“It was not a well-run process,” the White House official said.

On Jan. 20, the administration announced that religiously affiliated employers would have an extra year to comply with the birth control mandate. However, Catholic leaders perceived the grace period as an empty gesture and said they felt betrayed.

In the ensuing weeks, Biden continued to argue strenuously for a broader exemption. Deputy Chief of Staff Nancy-Ann DeParle, a top health-policy adviser, supported the idea, portraying it as an appropriate concession to Catholic nuns who had supported the administration — and opposed the bishops — during the health-care overhaul debate.

By late last week, the president had heard enough, the White House official said. He instructed his staff to resolve the matter within a week.

Staff writer David Nakamura contributed to this report.