It has been proposed, for various and well-meaning reasons, that death is something natural. This is not a mere difference in definition of nature that people have been using; people have meant it in very many different ways. It has been construed as natural because it occurs in nature, or because it is of the nature of a living thing, or because it is what one should want to do after a well-lived life. There have only ever been theological reasons offered against the naturality of death; death is a consequence of sin. Were the Garden of Eden not to be pierced by the corruption of evil, death would not have occurred.
I can’t say I’ve ever been wholly impressed by this view. For one, it misses the point of our still-remaining immortality. Death is not annihilation, it is just the ending of this part of our lives, and the beginning of a new one. This is so whether or not one has the fate of Heaven or Hell; the life of a human never ends. Would Adam and Eve have died if sin were to not occur? I’m not certain they wouldn’t have still died.
For all that, it still seems death is not something we should say is natural. Even if it is a result of life, that something is concomitant to a phenomena does not in itself make it natural. Approaching nature as the nature of things, that something dies does not show that death is the end of the thing.
We need to analyze the concept of nature as we apply it to beings. For it to be a nature is for something to have a certain aim, something that it seeks to accomplish. Inanimate matter has as its nature that of gravitation, the end of coming to rest in the locale of greatest mass. It can occur for times that a thing fails to do so; consider a comet endlessly whipping around the sun, never achieving rest. But we wouldn’t say that its continuing to careen around on a highly elliptical orbit is something it does; what it does is try and come to rest in the sun, which is why it returns to orbit. That it was impelled with such a velocity is something that happened to it. Therefore, that a thing moved is not of its own nature, at least not originally. It possesses the nature to be moved, at the very least, but that it is moved is not something of itself. This is the point to stick on.
So we see that a thing has something happen to it is not what we ought to understand as being its nature. It has the possibility of never achieving its nature, but this not because of the thing’s nature but because of circumstances external to it; thus not achieving its nature is something that happens to it, rather than something it does.
In this way we might understand that death is not something a living thing does. The end of a living thing is to live. That it is the end of a living thing to live is how we understand the form it possesses, how we interpret its parts as being concerned with the perpetuation of its life. An animal has a mouth that it can eat, a digestive tract that it might get nutrients from its food, legs that it might pursue food, skin that it might maintain bodily integrity, sexual organs that it might propagate others of its own kind, and so on. All this is concerned, ultimately, with its maintaining life. There is no organ or part of the animal that is concerned with achieving death. The animal doesn’t try to die. Death is something that happens to it.
As such, death is not something that comes of a thing’s nature. This is not to say that death might not yet happen to a living thing, but only to say that no thing has as its nature to die. Were it the nature of a living thing only to die, then it should never be a living thing, since its natures of living and dying would always be opposed to each other and it shouldn’t exist. But if its a living thing, then by definition its nature is to live. That’s what it is to be a living thing.

Jesus lived. One of the purposes (though certainly not the sole one) of Jesus’ life was to die. Thoughts?
That’s purpose, not nature.