What is this “separation of Church and State?” I don’t find that in my copy of the Constitution.
The Constitution contains three specific prohibitions relating to religion. First, no religious test may be applied as a prerequisite to a federal appointment.
Second, laws establishing a state religion are not allowed.
Third, the government may not interfere with the free exercise of your religion (within reason – you can’t claim that your religion requires you to smoke crack; or you can, but TS).
What you seem to be implying is if the government ever bans X or endorses Y, and X is also banned, or Y also endorsed, by some religious group, then that ban/endorsement somehow violates the Establishment Clause. But it is very, very clear that this is not necessarily (or not even most of the time) true. The government could have an independent, non-religious rationale for reaching the same conclusion as the Church (or, it could have an independent morally-founded rationale very similar to that of the Church). Many things forbidden by the Ten Commandmants are also forbidden by statutory law. Those statutes are cool constitutionally.
There are at least three independent moral/ethical/legal rationales for the death penalty.
The one you mention is “retribution.” Criminals must be punished. The death penalty as retribution theory says that no punishment other than death is sufficient for certain murderers.
A second rationale is deterrence. Deterrence comes in two flavors: specific deterrence (an executed murderer will most definitely not kill again); and general deterrence (e.g., I argue that would-be killers will be deterred from murder by the knowledge that the State routinely administers execution for those who murder).
The third rationale could be termed an “exemplary” rationale. The justice system needs to work, but it also needs to be seen to be working. This may seem to overlap with “general deterrence,” but it’s more than that – on this theory, even the most law abiding citizens, who would never even be within the realm of possible murderers, benefit from a highly-visible affirmation that the State is in control of marauders and that the justice system functions even in the most severe cases of wrongdoing. This is why we use the phrase “execute” – the State is literally executing, that is, putting into effect, the will of the jury (proxies for the people at large) and the judiciary.