Part of the reason it is more expensive in the US is because medical school is very costly in the US. This increases the financial risk of those aquiring a medical degree. Higher risks usually demands higher profits in the market.

Hence free university education could be a factor in reducing US health care costs.

Fundamentally you see how the free market approach to everything drives up costs everywhere.

Or more specifically drives up costs in industries/services which don’t naturally have functioning markets.

Free market approaches to health care found in Europe or Asia that works tend to be strongly regulated. E.g. max prices for procedures