On Friday the Obama administration filed its argument with the Supreme Court for why some Catholic groups should not be allowed to deny their employees contraception coverage, the latest salvo in an ongoing battle over how to treat those with religious beliefs under its new health law. While it remains unclear how the nation's highest court will resolve the issue, one thing is clear: while Americans back the idea of contraception coverage in general, they are divided over whether to compel businesses and nonprofits with religious objections to provide it.

The Affordable Care Act's mandate that employers cover contraception and related medications and procedures, such as the birth control pill, the morning-after pill and permanent measures such as tubal ligation, without a co-pay, has emerged as one of the law's most contentious provisions. It has prompted dozens of lawsuits from organizations that argue it violates their constitutional right to religious freedom. The Supreme Court will hear arguments on the issue this year.

On Tuesday Justice Sonia Sotomayor issued a stay at the request of an order of nuns from Colorado who said the rule violated their religious freedom, giving them a temporary reprieve from the requirement. The Catholic Church opposes artificial birth control.

The owners of some corporations have separately made the case that providing insurance that covers some contraceptives — or any contraceptives, according to some of the complaints — violates their religious freedom. The Supreme Court will decide that question later this year.

While the battle is playing out in the courts, the public remains torn over how to resolve it. Surveys show wide support for requiring birth control coverage in health insurance -- especially for publicly held corporations -- but significantly less support when religious objections are involved.

March 2012 Post-ABC national poll, for example, found 61 percent saying birth control should be covered by health insurance, with 50 percent backing this "strongly." But when these same people were asked whether the government should require such coverage when it is provided through religiously-affiliated employers who oppose birth control, support for the mandate dropped to 49 percent. About one in 10 respondents (11 percent to be exact) changed their mind when religious objections were brought to light, boosting opposition from 35 to 46 percent.

Other surveys find Americans making a similar case-by-case judgment. A CBS News/New York Times poll in December 2013 found 55 percent saying employers should be able to opt out of covering the full cost of birth control for female employees if they have religious or moral objections, compared to 37 percent who favored a requirement. By contrast, 66 percent supported the mandate when CBS/NYT first asked about the mandate without explicit mention of religious or moral objections of employers in February 2012. A Fall 2012 Public Religion Research Institute survey also found a range of support for a mandate -- 62 percent for publicly held corporations vs. 42 percent for churches who held religious objections.

The malleability of public opinion indicates Americans may be weighing competing values, including broad acceptance of birth control as a part of health coverage and a sensitivity to sincere religious objections.

Regardless of public attitudes, the administration is not backing down when it comes to its contraception coverage requirement. Nonprofits such as the nuns from Denver may opt out of the requirement simply by certifying that they have religious objections, Soliciator General Donald B. Verrilli Jr. wrote in a response to Sotomayor filed Friday morning.

“The employer-applicants here are eligible for religious accommodations set out in the regulations that exempt them from any requirement ‘to contract, arrange, pay, or refer for contraceptive coverage,’” Verrilli wrote. “They need only self-certify that they are non-profit organizations that hold themselves out as religious and have religious objections to providing coverage for contraceptive services.”