Saying it's the fault of religion doesn't help solve anything. It's individuals who decide to interpret the Bible in ways that harm women, just like other individuals can consider themselves Christian, or Muslim, or Jewish, and still support women's right to birth control.
And on what grounds are you going to claim that they should interpret their holy books differently?
There is only one valid, fundamental basis for making arguments one way or another, and that's reason. Religion is a systematic (albeit, potentially compartmentalized) rejection of reason.
Fortunately, going by reason, we can establish (with significant effort) that it is in everyone's interest to recognize certain rights for human beings, and we can institute governments to protect those rights. And, we can establish (with some effort) that hurting women is destructive and evil and malevolent.
Trying to argue from any other perspective is just saying, "It should be this way, just because."