Not really though. Like, if I am having a heart attack the correct answer to "Which hospital do I go to?" is always going to be "The closest one equipped to handle it". Well technically the one I get can to the fastest that can handle it but you see my point.
This kind of choice via competition would work for things like optometrists and GPs (assuming you live in an area with more than one general practice) but just plain doesn't for emergencies.
An because it's so obvious, people will want to be prepared for it. By having insurance (real insurance, not the bizarre "health insurance" construct we have today), or subscribing to some service like you do for when you car breaks down etc.
I'm not sure what exactly would happen, but I'm certain it won't be that people will just never think about this until they all, one by one, have a medical emergency and have to unexpectedly pick a hospital on the spot.
Because if hospital B isn't my choice but I'm closer to it when the emergency happens I am going to hospital B
In a free market, you'd also have choice via competition to drive that price down. Right now, hospitals can literally veto their competition by claiming "there isn't enough demand to justify another hospital in this area". So bing, no new hospital, no competition.
And of course, this assumes your provider is out to screw you at every turn. This turns out to not be the case for obvious reasons; your provider still wants your business in the future. In the case of a single hospital/provider (a la regulation) they've got your business no matter what.
Competition is a key force in markets, and regulation in the medical industry provides huge barriers to competition.