Are you talking about a potential situation where a Drugs-R-Us are using (for example) an American service, but directing them to store it on EU servers?
Or are you talking about Drugs-R-Us using a non-American service in general?
The internet however is not limited to US and EU. Criminals have always been using services all over the globe.
Ah, Americans. "Valid warrants and court orders" are national. Unless there are international agreements (of which there are many), whatever Bumf** PD, Oregon does should have no impact on what a Lithuanian server hosting company does. Of course, this is not how it works in real life and many non-Americans consider that abusive, and rightfully so.
FYI - this kind of extra-territoriality was what started WW1. An ethnic Serb Bosnian (Gavrilo Princip) shot the Austro-Hungarian heir apparent and as a result Austria-Hungary sent Serbia an ultimatum.
The ultimatum contained lots of harsh terms, which Serbia accepted, except for 1. The term they refused was to allow Austro-Hungarian police, prosecutors, judges to directly investigate, arrest, prosecute and imprison Gavrilo's possible collaborators. And the reason they refused that was because it's basically giving up sovereignty. Once you let another country send their police force investigate, arrest, prosecute and imprison whomever they want, they have full control. They can imprison your prime minister and you have no recourse except for starting a war.
So Serbia accepted that if it wanted to remain independent, it would have to accept fighting a war.
I have yet to see the EU ignore a valid judicial warrant except in the most extreme cases.
The point, however, is that you have to have a valid judicialwarrant and not some random-ass piece of paper generated by Adderall-addled sycophants of fascist South African nepobabies.