Can religious institutions discriminate when selecting their own clergy?

Question

According to the U.S. Equal Employment Opportunity Commission, it is illegal to discriminate against a current employee or candidate based on race, color, religion, sex (including gender identity, sexual orientation, and pregnancy), national origin, age (40 or older), disability or genetic information.

Does this apply to religious institutions selecting their own clergy? I'm going to guess that it's allowed if there is a religions reason (e.g. the Roman Catholic church has a religious restriction forbidding women from being priests). But if there is not a religious reason, would they be permitted to discriminate?

For instance, could a Protestant church that does not have any religious restriction regarding the race of clergy still choose to discriminate based on race for their own reasons (perhaps they only want to hire a black pastor, or say that they will reject candidates who are black pastors)?

Or are religious institutions given some sort of protection regarding their hiring practices?

Answer 1

This is known as the "ministerial exception". Because the Free Exercise and Estalishment clauses of the First Amendment prohibit the government from interfering with religion, the government cannot override a doctrine that contradicts the teachings of a religion (so women and gays cannot sue the Catholic church for not being hirable as priests).

In Hosanna-Tabor v. EEOC, an individual taught classes and led prayer at a religous school, but was fired ultimately due to a disability (narcolepsy). The Lutheran church does not have any known doctrine condemning narcolepsy: but it was unanimously ruled that "the Establishment Clause prevents the Government from appointing ministers, and the Free Exercise Clause prevents it from interfering with the freedom of religious groups to select their own". Thus the church was legally permitted to fire the individual due to her disability.