True, Europe does not have one healthcare system. And yet, thanks to the EHIC (European Health Insurance Card) [0], citizens of the 27 countries in the EU, Switzerland, the UK, and the EEA (Iceland, Liechtenstein and Norway), can use the healthcare systems from any other of the 31/28 [1] participant countries if they need healthcare on a temporary visit under the same conditions as the local resident citizens. The expenses incurred are reimbursed to the system providing the healthcare by the healthcare system of the citizen who is receiving it. This is done transparently to the patient, all they need to do is provide their EHIC card.

This means that each of these 32 different healthcare systems, with their own particularities (e.g. free at the point of use vs. copay), can be considered one single universal healthcare system.

If 32 different countries, with their own legal systems, legislation, languages, and healthcare system design (insurance based or state funded) can cover approximately 500 million citizens, then 50 US states under one single federal government should have no problem doing the same for their 333 million citizens.

[0] https://en.wikipedia.org/wiki/European_Health_Insurance_Card

[1] In the case of the UK, the EHIC only reciprocal with EU and Switzerland, not the EEA.