Each country in the EU has its own health care system, but in most senses they are regulated EU wide, and reciprocate with each other - if a French person needs urgent care in Italy, he will get it, and the distinct single payer systems will do the accounting. Said French person can also legally move to Italy, and will be covered by Italian single payer system from day one.

The EU has only one medical "FDA"-like authority, which is - in fact - a huge problem for the UK post Brexit because they have to set one up for themselves now.

The EU has, effectively, a EU-wide health care system. It is more federated than the US, by virtue of only recently become a union. It is also less uniform because of different history, tradition, and languages. These are things that make it harder to build than a US-wide, and yet, it works and is significantly more efficient than the US, while catering to more people.

Of course the existing US-wide health care system is not equivalent; it is dysfunctional. You claim, but fail to demonstrate, that a functional one cannot be built, despite example in the EU that faces harder objective conditions.