In the '60s, radioactive [anything] could lead to anything else. Nowadays, it's genetically engineered [anything] that gets the same result. If Parker's going to gain spider-like powers for the sake of the plot, then the web-generation thing is a reasonable side-effect. It makes no scientific sense, of course, but that's so far beside the point as to be not worth considering.

Though I read an article on a 'movie science' site which calculated exactly how much thread Mr Parker might realistically produce. Based on the diameter of the thread, I think they worked out he could produce perhaps 200m of thread before he'd have exhausted the equivalent of his entire body volume.

Moral of the story: if you want rigid adherence to scientific reality, don't watch Spider-Man, X-Men, or any other comic-based film. They're meant to be fun, not educational - though with that kind of story, internal consistency is a plus.

:edit: Oh, and much glee at the Futurama reference in the thread title <3