There are exceptions to the Title VII prohibition, which "does not apply to discrimination by a religious organization on the basis of religion in hiring and discharge. The exemption applies to an organization whose 'purpose and character are primarily religious.'" In all other respects, a religious organization is bound by the law that everyone else must follow. The EEOC defines "religion" as "includ[ing] moral or ethical beliefs as to right and wrong that are sincerely held with the strength of traditional religious views", but that does not seem to encompass metaphysical beliefs (e.g. about reincarnation, who created the universe...). They also state that "Religious discrimination also includes discrimination against someone because s/he is an atheist". The wording there is of some significance, since it does not say that an atheist organization is a "religious organization" which would be entitled to discriminate in favor of atheists in hiring practices. The statutory definitions section of the law tells us that "religion"
includes all aspects of religious observance and practice, as well as
belief, unless an employer demonstrates that he is unable to
reasonably accommodate to an employee’s or prospective employee’s
religious observance or practice without undue hardship on the conduct
of the employer’s business
which doesn't actually support the commission's decision to extend coverage to atheist employees. That interpretation comes from post-statutory case law, such as Shapolia v. Los Alamos Nat'l Lab., 773 F. Supp. 304, 305.
What is unclear is what a "religious organization" is. The exemption comes from 42 USC 2000e–1(a):
This subchapter shall not apply to an employer with respect to the
employment of aliens outside any State, or to a religious corporation,
association, educational institution, or society with respect to the
employment of individuals of a particular religion to perform work
connected with the carrying on by such corporation, association,
educational institution, or society of its activities
but the definitions do not clearly state that an organization dedicates to denying religious beliefs is legally subsumed under the exemption. Billy Graham would be, dunno about an atheist organization.