This is a matter of opinion, of course. The government is allowed to violate people's right to free exercise when it has a "compelling interest" in doing so. That means, for example, that Amish people can be forced to pay taxes even if it goes against their beliefs. It means that people who follow American Indian religions can be prevented from taking drugs as part of that religion. It means that polygamy can be banned.
So the question is whether the government has a compelling interest in requiring religious institutions to provide birth control as part of their medical programs. You could say that the government does have such an interest because it needs to protect women's rights to have access to contraception. However, there are many who would argue against this point of view.