Countries hosting the data centres can make it illegal to allow access from outside their area/EU... or specifically to US entities along with making it illegal to move any data out without customer/local gov approval... This isn't rocket science. The company cannot do business if it doesn't follow the law. There are laws like this in places already. The company's local subsidiary tells the American company to politely pound sand and the American company says sorry, we tried, but do not have the capability to do as asked.
An American company will always follow US law, no matter the local laws.
GDPR give exemption for foreign government for "national security", "important reasons of public interest" or "law enforcement", whatever that meant.
America has become China in the eyes of the world.
Everyone banned Huawei products despite the ability to pass laws saying Huawei must respect data sovereignty. They didn't ban US firms, because unlike China the USA was championing the rule of law at the time. Data sovereignty only works if the USA respects the laws of other countries, even though, just like China, they could coerce / bribe citizens and firms to bypass them. Such activity would be largely undetectable. Who is going to know if someone peeked at a secret document stored in Azure? There was a huge amount of trust involved in the arrangement.
The USA has now denounced the rule of law, is withdrawing the the institutions set up to champion it, and has shut down the ICCC's access to some services. The trust has gone.