I noticed yesterday that Republicans ran away from further votes on the issue of birth control. It was pretty obvious that their position, which started as a paean to religious freedom but started to sound suspiciously like conservatives just didn’t believe in birth control – certainly not as a preventive health measure, was hurting them with the public. But we didn’t know how much until a new poll secured the data.

The poll, conducted by Greenberg Quinlan Rosner Research for the progressive groups EMILYs List and the Planned Parenthood Action Fund, showed that Democrats hold a wide lead over Republicans on issues like access to birth control (56-18), women’s health (46-28) and abortion (42-31). In addition, voters strongly opposed the Blunt amendment, which would have allowed employers to deny preventive health care coverage as part of their insurance plans if they raised a moral objection, by a 60-40 margin. But more importantly, 48% of respondents said they would be less likely to vote for a candidate if they supported the Blunt amendment. That’s a significant number that puts metrics to the belief that the debate hurt Republicans generally.

What’s more, GQRR ran an experiment revealing that the birth control issue actually moves numbers:

Access to birth control has the potential to impact actual races. As a starting point, in this battleground, a generic Republican leads a generic Democrat by 5 points (47 percent to 42 percent). In a split experiment, we tested a generic informed match-up between a Democrat and a Republican with half the sample; in this exercise, the Democrat continues to trail by 6 points (47 percent to 53 percent). The other split received the same information with accompanying information about birth control. In this match up, the candidates are tied at 48 percent.

On the politics, it does appear that Republicans are right to back off – their position on birth control was severely damaging their party brand. On the policy, it’s worth noting that there is active litigation on the rule promulgated by the Health and Human Services Department that mandates contraception coverage as part of the preventive health services menu, free without a co-pay, with some exemptions and accommodations for religiously affiliated institutions. Conservative and religious groups have filed eight separate lawsuits protesting the rule. The rule is also part of the broader Affordable Care Act, which the Supreme Court could rule unconstitutional later this year. The conservative groups suing over the birth control rule do not have precedent on their side:

To win that argument, they will need to clear a major legal hurdle: A landmark 1990 decision in a case called Employment Division v. Smith, in which the Supreme Court found that if a law is “neutral and generally applicable” — meaning that it is not specifically targeted against any religious group — individuals must comply with it even when doing so imposes a burden on their free exercise of religion.

Writing for the majority in that case, Justice Antonin Scalia — a conservative justice known for his strong identification with the Catholic Church — found that to allow otherwise “would be courting anarchy” by making “the professed doctrines of religious belief superior to the law of the land and in effect to permit every citizen to become a law unto himself.”

Nonetheless, with the politicians running screaming the courts are probably the last area of potential relief for those opposed to this rule. And they don’t look likely to succeed there.