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President Obama on Friday backed off a plan requiring religious organizations to provide free contraceptive coverage, shifting the responsibility to insurers in a move he claimed would preserve religious liberty as well as "basic fairness" for women who want the coverage.

The announcement comes after a contentious week of debate in Washington and across the country about the rule, which stems from the federal health care overhaul. Religious groups, particularly Catholic organizations, decried the policy, while lawmakers on both sides of the aisle charged it was an infringement on religious freedom.

But supporters defended the intent of the policy -- to make sure women at all income levels have access to affordable birth control no matter where they work.

The change did not appear to satisfy some of the toughest critics. Rep. Jim Jordan, R-Ohio, chairman of the Republican Study Committee, called it a "fig leaf, not a compromise." A spokesman for House Speaker John Boehner said the religious community is "not yet convinced" the revised rule is not an "attack on religious freedom," and said the House would continue to work toward a "legislative solution."

More than three-dozen religious scholars and leaders of different faiths also put out a statement Friday afternoon calling the change "unacceptable." Academics and officials from the University of Notre Dame, Harvard University and other institutions argued that the shift "changes nothing of moral substance," since it still "coerces" religious groups to offer policies that will, regardless of whether the groups are paying for it, offer contraceptive coverage.

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But Obama said the issue should not be treated as a "political wedge."

"Under the rule, women will still have access to free preventive care that includes contraceptive services, no matter where they work -- that core principle remains," Obama said. But he added, "Religious organizations won't have to pay for these services, and no religious institution will have to provide these services directly."

The change would allow religious organizations to refuse to cover contraceptive care. It would also require insurers to offer a plan that does not include contraceptive care in their contracts with nonprofit religious groups. But the insurers would be required to make contraception available free of charge to women anyway.

"We think this is a very workable solution," Health and Human Services Secretary Kathleen Sebelius told Fox News, adding that it respects "religious objections."

Advisers stressed that the announcement was an "accommodation" and not a "compromise." Administration officials say the many media accounts of the internal deliberations have been "over-dramatized." Obama said the debate over the issue effectively sped up a discussion that was already happening on how to proceed.

The move came after several prominent Democratic lawmakers urged the Obama administration to reconsider its position. Vice President Biden said in an interview a day earlier that he was "determined" to work out the dispute.

Biden was among a string of Catholic administration officials who reportedly voiced concerns with the policy as it was being hashed out. Obama announced the rule in January, requiring organizations to provide birth control coverage without a co-pay for employees. The rule exempts churches and houses of worship but would apply to other religious organizations in 2013.

Catholic leaders vigorously opposed the policy. In Congress, Republicans threatened to overturn the rule if the administration did not change course. The issue also took center stage at the Conservative Political Action Conference in Washington this week, and on the Republican presidential campaign trail.

Still, Obama had rallied Democratic allies to his side in defense of the policy and its intent to expand access for women to contraceptive services. Claims that the change does not mark a compromise may indicate an unwillingness by the administration to be seen as backpedaling on the issue.

The administration circulated statements of support Friday from both Planned Parenthood and the Catholic Health Association of the United States. Former Virginia Gov. Tim Kaine, a Democrat who had been critical of the original plan, also praised the change.

The U.S. Conference of Catholic Bishops called it a "first step in the right direction" but said it was reserving judgment.

But other officials expressed disappointment. Sen. Orrin Hatch, R-Utah, called for a full exemption for religious groups.

"This is about religious freedom, and anything short of a full exemption is no compromise. The White House has shown time and time again that nothing will stand in the way of politics, and their actions on this mandate reflect that," he said in a statement. "The backlash surrounding the White House's decision to force religious institutions to act against their beliefs lays that fundamental fact that the president's health law is unconstitutional to its very core."

Fox News' Ed Henry contributed to this report.