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.
The copyright law introduced during the invention of the printing press is nothing like the copyright law we know today.
Back then ideas were still believed to be free so the point of copyright law was just a short term reward for the author. A bit like how patents are supposed to work.
I would normally post some citations here (like a famous quote about copyright from one of the British monarchs) but on phone about to drop kids off at school so apologies there.
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.