@vivify said
Should Catholic doctors be allowed to deny people birth control because of their faith?
Patients shouldn't need to visit a doctor to get birth control. They should be able to buy contraceptive pills from a pharmacist without prescription.
This is the foolish rabbit hole of allowing unscientific beliefs like faith dictate medical treatment or politics.
Christian belief is one motive for anti-abortion sentiment, but not all pro-lifers are religious. I don't see why someone who believes that an unborn child has a right to life, whether on secular or religious grounds, should be required to act against their conscience.
When abortion was legalised in West Germany in 1974, the Federal Constitutional Court struck the law down because it infringed on the right to life of the unborn child. There was no reference to religious dogma in the decision; the judgement was reached on human rights grounds. It found that:
http://groups.csail.mit.edu/mac/users/rauch/germandecision/german_abortion_decision2.html
"The State's duty to protect forbids not only direct state attacks against life developing itself, but also requires the state to protect and foster this life", that "the obligation of the state to protect the life developing itself exists, even against the mother," and that "the protection of life of the child en ventre sa mere takes precedence as a matter of principle for the entire duration of the pregnancy over the right of the pregnant woman to self-determination."
The German court considered the reasoning of the then very recent Roe v. Wade judgement, and explicitly rejected it on the grounds that it was inconsistent with constitutional guarantees of the right to life.
It's also one of the many consequences of Europe becoming increasingly right-wing.
In the same period when laws restricting abortion were tightened in Hungary and Poland (i.e., within the last five years), they were relaxed in Ireland, Cyprus and the Netherlands. The picture's much more complicated than you suggest.