The title sounds like a bigger deal than it really is. The reality is some "closely held" companies can squirm out of providing contraception on religious grounds, but it only referred to morning-after pills, which could be ideologically argued are along the lines of abortion. This ruling didn't have effect over preventative contraceptives.
This won't go much further on a wider scale. A few privately held companies with religious owners might do this on ideological grounds, but public-limited companies won't be able to, and really no large corporation in their right mind would want to go down this road because the PR backlash is going to massively outweigh the minuscule financial savings from not providing certain contraceptives as part of their insurance plan.