You cannot police every website on the Internet, but you certainly can scrap the Internet. It is easy to forget how much regulation surrounds the physical infrastructure of the Internet and how easily the government could just shut it all down. A key property of the Internet is that there is a single public IP address space; there is no technical reason why the address space could not be divided into "client" and "server" addresses, with only "servers" being allowed to host applications, and we are already halfway there with NAT (IPv6 does not help either, as it could easily be fragmented and we already have things like ULA). It would be easy to require a special license to receive a "server IP address" and I can see the EU doing exactly that based on their recent pattern of behavior.

Europe has a long history of doing such things when confronted with new, disruptive technologies: the effort to license printing presses in various European countries is what eventually led to copyright law as we know it today.