I don't know how the politics will play out, although I doubt that the backlash will really impact the Presidential election to any great degree.
But I believe that if Catholic hospitals and doctors are going to accept taxpayer dollars in return for their services, they shouldn't be allowed to refuse to provide contraceptives to patients who request them.
No Catholic hospital or doctor has to take any federal money if they don't want to; but if they take the money, they should have to play by the same rules that apply to everyone else.
This is different from the abortion situation because federal law already prohibits the use of taxpayer money to fund abortion services. The Catholic hierarchy knows that they will never get a law banning the use of taxpayer money to provide contraceptives through the Congress, so they are raising this phony religiou freedom argument instead.