There are some medical treatments which are sorta-necessary-but-not-right-away. Um, like hip replacements. Or maybe some types of post-accident reconstructive surgery. Or...I don't know, there are probably more, but I can't come up with all of that many. For those the magic of price competition could work...but, so what? It's just not an important of a class of medical procedures.
Generally, the best comparison is probably shopping for an auto mechanic. You probably can get a couple of estimates, but each one is going to take an hour of your life at least, unless your car needs to be towed in which case one more estimate is also going to set you back another hundred or more bucks. What you care about, really, is less the price than about whether or not you can trust the auto person to do the job well and that they aren't either incompetent or lying about how to fix your car. Knocking a few bucks an hour off the labor charge, or finding a deal for you on aftermarket parts, is of some value but far less than the value of being sure they're really going to do your job right.
And most people probably need their car to be fixed fairly quickly. Can get by for a few days without it, but...it's somewhat pressing. The point I'm getting at is, for all the reasons shopping for a mechanic is actually quite problematic, shopping for health care is a lot more problematic. Especially if you've been just towed to the hospital in an ambulance.