Contraception controversy consumes D.C., campaign House Democrats urge Obama not to back down in the contraception dispute A pending rule says religiously affiliated employers must provide full contraception coverage Conservative leaders argue the rule violates religious freedom Democrats note Massachusetts had a similar rule when Mitt Romney was governor WASHINGTON (CNN) -- A group of rank-and-file House Democrats urged President Barack Obama on Thursday not to abandon a pending federal rule requiring religiously affiliated employers to provide full contraception coverage to women -- one day after hints emerged of a possible compromise between the White House and conservative religious critics. "It is time for the extreme right wing to stop playing football with women's health," said Rep. Nita Lowey, D-New York. We "stand in solidarity with American women who have waited decades" for fairness in health coverage. Lowey said any retreat on the issue would be "outrageous." The question of whether institutions with religious ties should be forced to offer insurance plans covering birth control and the so-called "morning after pill," among other things, hits a number of political hot buttons. Liberal groups defend the requirement on grounds of gender equality in health care; conservatives consider it a violation of the First Amendment and an infringement on religious liberty. Millions of American Catholics have found themselves squarely in the middle of the controversy. Catholic teaching firmly opposes abortion and the use of contraceptives -- a point repeatedly stressed by numerous church leaders over the past several days. Polls, however, show a slight majority of U.S. Catholics actually favor the requirement. While churches themselves are exempt from the rule, hospitals and schools with religious affiliations would have to comply. The new policy is set to go into effect on August 1, though religious groups will have a year-long extension to implement the rule. Republican leaders have blasted the administration's decision, raising the issue's profile on both Capitol Hill and the presidential campaign trail. Some political analysts think the controversy will cost Obama votes in politically critical states like Pennsylvania and Ohio in November, while others insist it will ultimately hurt Republicans with suburban women. Bloomberg reported Wednesday that there was a deep internal administration split on the matter. Vice President Joe Biden and former White House Chief of Staff Bill Daley warned Obama about the possibility of negative political repercussions in swing states, while several female aides -- including Health and Human Secretary Kathleen Sebelius -- urged the president to move forward with the rule, Bloomberg said. White House Press Secretary Jay Carney denied the report, though he declined to offer any details. Two senior administration sources, meanwhile, told CNN Wednesday that White House policymakers are searching for a middle ground that would ensure women can receive full coverage without forcing Catholic charities, hospitals and other institutions to pay for it. The administration is particularly interested in the Hawaii model, in which female employees of religious institutions can purchase contraceptive coverage directly from the insurer at the same price offered to employees of all other employers. Policymakers also are examining laws in 28 states that have similar coverage requirements, sources said. Another possible solution, one source said, would be legislation allowing women employed by religiously affiliated employers to get contraceptive insurance from the exchanges created under Obama's sweeping health care reform, rather than from their employer's insurer. Sources familiar with White House thinking said the administration is trying to win over progressive Catholics, and is convinced approval from conservative Catholics is out of reach. New York Archbishop Timothy Dolan, head of the U.S. Conference of Catholic Bishops, said Thursday that the decision to require full contraception coverage was a "terribly misguided judgment." "We can't compromise on principle," Dolan told "CBS This Morning." That's almost rewarding bad behavior. While the regulations have caused a firestorm of criticism, a new study released by the Public Religion Research Institute shows the majority of Catholics support the administration's plan. Nearly 6 out of 10 Catholics think employers should be required to provide this kind of insurance coverage. A slight majority of Catholic voters -- 52% -- back the measure. On Wednesday, House Speaker John Boehner, R-Ohio, called the policy an "unambiguous attack on religious freedom" and announced his chamber would pursue legislative action to prevent the rule from going into effect. "If the president does not reverse the department's attack on religious freedom, then the Congress, acting on behalf of the American people and the Constitution that we're sworn to uphold and defend, must," Boehner said on the House floor. On the presidential campaign trail, GOP frontrunner Mitt Romney has pledged to eliminate the rule on his first day in office. Both the White House and Romney's Republican opponents, however, have noted a Massachusetts law in effect while Romney was governor that required hospitals -- including Catholic ones -- to provide emergency contraception to rape victims. It's ironic for Romney to criticize "the president for pursuing a policy that is virtually identical to the one that was in place when he was governor of Massachusetts," Carney said Wednesday. Romney, in turn, said Carney needs to "check his history." In 2005 then-Gov. Romney vetoed a bill that would have required all hospitals -- including Catholic hospitals -- to provide emergency contraception. The heavily Democratic state legislature overrode his veto. According to news reports at the time, Romney initially said his administration would not enforce the law at Catholic hospitals. But he later reversed course, saying all hospitals would have to supply the morning-after pill. Romney was quoted at the time as saying "my personal view in my heart of hearts is that people who are subject to rape should have the option of having emergency contraception or emergency contraception information." "I worked very hard to get the legislature to remove all of the mandated coverages, including contraception," Romney told reporters Wednesday. This "was a provision that got there before I did, and it was one that I fought to remove." The Democratic National Committee has scheduled a Thursday afternoon news conference featuring two Massachusetts Democrats who insist Romney never made a serious push as governor to repeal mandatory contraception coverage. CNN's Dan Gilgoff, Jessica Yellin, Brianna Keilar, and Dana Bash contributed to this report