IIS is still quite widely used for many web sites. Netcraft's latest survey results suggest just slightly under 30% of all web sites are served by it. So that's some closed-source software powering a significant portion of the web, at least.
And it's quite safe to say that those IIS installations are running on some variant of Windows, which is yet another generally closed-source software system. Then there are other non-web services (DNS, FTP, and so on) handled by such Windows systems.
There are still a surprising number of proprietary UNIX systems out there in production, on the public Internet. We're talking HP-UX, AIX, UnixWare, and even BSD/OS in some cases. There are many behind the scenes, indirectly supporting web sites and other publically-accessible services.
There is a lot of other networking gear that runs proprietary software, too.
Is open source software important to the Internet? Absolutely. Is it the "100%" you're claiming? Absolutely not.