The US has a market for private healthcare, but healthcare will never be a free market. It's the antithesis of a free market. Intrinsic inequality of bargaining power and information asymmetry are the biggest reasons for this.
This point is usually brought up in regards to healthcare emergencies. But emergencies only make up a small fraction of healthcare spending[1].
Maybe people have less bargaining power because of agreements with providers/insurers. That's a barrier to an effective market with price sensitivity, but it's a contingent one, not a mathematically inevitable one. For much of our healthcare spending the balance of bargaining power could look like it does for food.
1: https://www.politifact.com/factchecks/2013/oct/28/nick-gille...