Two new polls today shed some light on this question. The first one, from the Public Religion Research Institute, asked if all employers should be required to offer healthcare plans that cover contraception:
All Americans: broad agreement, 55%-40%
Catholics: broad agreement, 58%-37%
But maybe respondents weren't specifically primed to think that some employers are churches that have theological objections to birth control. So the second survey asked the general question first (getting similar results to the PRRI survey) and then asked specifically if Catholic hospitals and universities should be included:
All Americans: broad agreement, 57%-39%
Catholics: broad agreement, 53%-45%
In both cases, the numbers are much higher for Democrats and Independents. It's really only Republicans who object much, which strongly suggests that most of the objection is rooted in ideology, not religious conscience.