Which is why a virus would have to be truly massive to actually 'destroy' the internet.
Wouldn't such a virus take an obscenely long time to get uploaded and around the place? Long enough, perhaps, that it would be caught and terminated before anything major happened?
Viruses are fast. Really, really fast. When they want to be.
Warhol worms, for example, are defined by infecting 95% of vulnerable hosts in 15 minutes. Since most of the vulnerable hosts are people with no idea about computer security at all, it's unlikely that even if that were 15 hours or 15 days a significant number of the systems could be immunized. Conficker, for example, used a vulnerability that had had a patch available for months before it was released.
Those viruses have a tendency to overload things trying to spread themselves, so they can bring things to a horrible grinding halt (both the virus and the internet) if the creators aren't careful. They've been getting better about that recently, though, and some of them spread themselves extremely slowly and carefully to avoid detection.
How can a virus "destroy" the internet? The short answer is that it can't, but the longer answer is that viruses can use the large amount of bandwidth that they have between all of their infected hosts to saturate a single site (DDoS attack) or slow down traffic for the internet in general (what happens when a virus doesn't spread well). That doesn't actually destroy the internet, but it can slow it down noticeably, normally until ISPs can start blocking infected computers.