Most countries have their own, but they're mostly for copyright infringement, not GDPR violations.
Even that isn't universal, Poland's system only affects unregistered gambling websites for example, and I've seen quite a few ISPs that don't even bother enforcing it, even though they're legally supposed to do so.
There's nothing (except talking to your government) that the EU could realistically do at this time to block a website, and I genuinely don't know how receptive the US would be to these arguments.
The largest "hook" the EU has is that most companies that provide services to you, whether that'd be payment processors, hosting services or ad networks, (still) want to maintain good relations with them and don't want to burn bridges, and I don't believe it's beneath the European commission to put pressure on those to make your life difficult.
If you take direct payments, you'll probably have a talk to your card acquirer on how they feel about EU clients and how you deal with them. They might decide to not do any business with you depending on that.
In these above examples, I think having wider scale actually helps negotiating better terms instead of getting kicked out at the first occasion.
If you're in a hypothetical country that the EU has no relevant treaties with, the EU has no power over you. They might claim that EU laws are extraterritorial and affect everybody who dares to appear on the internet without blocking EU citizens, but that claim can't be enforced in such a country.