These websites would logically have to move to EU based servers in order to be subjected to the kind of regulation that EU citizens want. I think it could work.
It seems like this would only work if EU citizens were required to use the EU browser and nothing else. At what point does protection turn into control?
Presumably, EU citizens want EU regulations for privacy. I'm suggesting that an EU browser merely provides a warning when a non-EU regulated site is being visited. So, the user is in control and it avoids the scenario where the internet becomes effectively regulated by the union of all regulations from all jurisdictions.