Wrong question. The Constitution prohibits the government, whether federal (5th Amendment) or state (14th Amendment) from depriving a "person" of life (or property) without due process. But a murder (or theft) committed by a private person against another private person does not violate the constitution; murder is only a state-law crime. Many states permit killings (as self-defense) which would be murder in other states (see, for example, the wide differences among states as to whether or not you can legally kill an unarmed thief).
My counterargument to your argument would be this:
It may currently be true that the definition of person varies based on the context, but I don't believe that it should. It should be easy to specify what a person is, so that regardless of the situation, we all know what we are talking about when we use the word "person".
So even if the Constitution is not currently concerned with private crimes such as murder and theft (and I don't believe that it ever should be directly), it seems that it would be a good idea for the Constitution to clearly define what is meant by a "person".
Then if individual states want to craft laws that allow some persons to be killed in certain situations, they would at least have to own up to the fact that that is indeed what they are doing.
The Constitution could merely define a human being as being so from the point of conception without even mentioning abortion or any other consideration. Then all of the states would have to recraft their laws accordingly. They could still make abortion legal, but at least they would have to admit it was the killing of an innocent and non-threatening person.