Note: I will use Romania as an example in my post, but the question is general (EU level).
Environment-related infringement procedures are something rather usual in the EU. Just to pick an example, Romania was called to close and rehabilitate illegal landfills:
The timeline for the closure and rehabilitation of the 15 remaining landfills is uncertain as for most of these landfills the closure works have not yet started. The Commission is therefore addressing a letter of formal notice to Romania. Romania now has two months to respond to the letter, otherwise the Commission may decide to issue a reasoned opinion.
On the other hand, I am unable to find any similar procedures related to health system quality. A few of the EU countries have a high preventable and treatable mortality rate (RO, BG, LV) and I think an important factor is a health system quality.
In the specific case of Romania, I would find this document that recognizes the health system poor performance:
Per capita health spending of EUR 814 in 2015 is the lowest in the EU, and under a third of the EU average
The only thing loosely related to what I want to find is this article which mentions infrigements related to cross-border care:
We have learned that some infringement procedures – launched by the European Commission with regard to cross-border care, or payment for such care – can be seen as challenges to the basic objectives and functions of national health care systems and to the governance role of national health authorities.
Since people's health is arguably at least as important as the environment, I would expect some "pressure" to also be applied in the health area.
Question: Did the European Commission start an infringement procedure related to the quality of one or multiple EU members health systems?