Yes it does, according to this report from the LSE. Not exactly a hotbed of free-market thinking, but here is what they conclude in their (peer-reviewed) report::
"Using AMI mortality as a quality indicator, we find that mortality fell more quickly (i.e. quality improved) for patients living in more competitive markets after the introduction of hospital competition in January 2006. Our results suggest that hospital competition in markets with fixed prices can lead to improvements in clinical quality."