So did Charlotte die because of Wilbur?

I never thought of this before, but there's that scene where Wilbur begs Charlotte to let a fly go, and she does. Then she says how she really could have used the nourishment because she's about to lay her eggs. Then, sure enough, right after she does, she dies.

Is that supposed to be the reason she wasted away after giving birth? Like it's so taxing on the body that if you're not in really good health, it'll kill you? Or was it somehow tied to the webs she made for Wilbur, like that took a lot out of her system to the point that one more extreme activity (giving birth) killed her?

I always felt like it would have made more sense, dramatically, if Charlotte sacrificed herself in some way, even only indirectly, for Wilbur. But they don't make that very clear in the movie. I admit I haven't read the book, so maybe they explain it there.