The conflicts between religious beliefs and secular laws are numerous. Our taxes pay for alcohol production and sales regulation enforcement, but alcohol consumption is prohibited under Mormonism. Businesses are open on Saturday, but working on Saturday is prohibited by Judaism. Tattooing regulation is funded by taxes, but Islam prohibits tattoos. Regulation of hair cutting businesses is funded by taxes, but cutting hair is prohibited by Sikhism. Meat production and sales regulation is funded by taxes, but meat consumption is prohibited by Hinduism.
The empirical evidence demonstrates that availability of contraception correlates positively with better health outcomes for women and children such that the additional cost of covering contraception in health plans is more than offset by the overall health benefits. So government has an obligation to require that employer health insurance covers contraception. But some religions prohibit contraception, so isn't such a law a violation of the Free Exercise clause? The Catholic church, among others, argues that it is whenever the employer is affiliated with their church.
Notre Dame and Saint Mary's, to give two examples, are Catholic universities. Religious corporations, associations, educational institutions, or societies are exempt from the federal laws that EEOC enforces when it comes to the employment of individuals based on their particular religion. So can government require that Notre Dame and Saint Mary's subsidize health insurance that covers contraception?
No statement of faith is required to attend or teach at Notre Dame or Saint Mary's. There is no doctrinal control over what is taught in the classroom. Faculty, staff and students are encouraged to seek the truth in whatever form it takes, including that of scientific research.
Institutions like Notre Dame and Saint Mary's must make a choice. Either they commit to being modern universities and accept all that entails, or they commit to being primarily Catholic. They cannot plead special exemptions from generally applicable laws on the grounds that they are a religious institution on the one hand while placing no religious restrictions on the beliefs of employees and students on the other hand. Religious institutions have no free exercise right to impose their religious beliefs on their employees against the will of those employees when those institutions do not require their employees to profess or practice those religious beliefs in the first place.