There might not be a database covering all countries of Europe (for instance the non-democratic country Belarus does not publish much information). So if you need more data that OpenStreetMap has, you will probably need to go country-by-country. Below is the data for France.
The French government has created a website showing the list of all 4307 hospitals and clinics in France:
Click on "Voir la liste des établissements" to get the full list.
You will need to scrape that. Each hospital name has a link to its detailed information page:
I have just sent them a message asking for a dump of the data. They replied to me saying that all of their data comes from the FINESS database, which is also maintained by the French government, and available online "for private use" and unfortunately does not have dump downloads, so that would mean scraping that: