Writing in The Federalist, Jay Richards criticizes the military for denying service members religious exemption from the Covid vaccine. He argues that doing so interferes with the First Amendment rights of Roman Catholics and others opposed on religious grounds to any and all abortions, since the vaccines were tested on aborted fetal tissue.
On the other hand, service in the military is not an inalienable right. For example, we do not permit pacifists to serve, those who refuse to use force to defend themselves and the nation.
Military service makes demands on, and requires obedience from, its members. These demands sometimes contravene the dogma of this or that religious belief and interfere with constitutionally protected rights. It demands sacrifices members of the all-volunteer military choose to make willingly, or they find another line of work.
The military observes most of its members who are Catholic have been vaccinated, as required. Thus it finds the refusal of the relatively small number claiming religious exemption to be disingenuous and likely based on something other than demands of faith. In this case I side with the military, which needs healthy troops.