It varies a lot, there really isn't any common customs in Europe in that respect so creating a list might be the best way to answer the question. Complimentary water is definitely not limited to Finland or Scandinavia.
Countries where serving tap water is required by law:
- UK (source provided by @Traveller). Legally required in all premises licensed to serve alcohol. Most other cafes and restaurants offer free water too (but not all).
- France (source provided by
@Franck Dernoncourt). It's very common to just bring a pitcher of water on the table of anybody who orders food or at least to offer one even if you didn't ask for it or ordered other drinks.
- Spain (source provided by @Erik)
Countries where serving complimentary water is very common:
Countries where complimentary tap water is not unheard of but not universal:
Countries where serving free tap water is unusual:
- Netherlands. In fact, it's not uncommon to ask for “Spa rood” or “Spa blauw” instead of water generally… and waiters will simply bring another brand without warning or confirmation if they don't carry Spa. But asking for 'kraanwater' (tap water) is common as well, sometimes you get it, sometimes you are told to order mineral water instead.
- Germany. Fizzy drinks are very popular and water is sparkling by default. In the bigger cities it it more common to get free water than in the countryside. And altough most places don't offer it themselves, if asked they will either refuse and point to bottled water or just give it to you for free (and the places where you get some for free if you ask are not few). It is very uncommon to get charged for tap water.
- Belgium. When asking flat/non-sparkling water, it will be of a bottle.
In those countries, patrons will almost universally order paid drinks with a meal and the issue doesn't come up. In some cases, restaurants will flat-out refuse even if you insist, in others, they might comply but will be surprised or annoyed because it is very unusual.
Conversely, even in countries where serving complimentary water is mandated by law, you will find places where it is not available or simply not done. Examples include mountain huts and upscale restaurants. I don't know if they would push back if you insist that they are required by law to serve tap water but in all Michelin starred restaurants I have been to, patrons are clearly expected to order paid drinks. Water filtered and possibly carbonated on the premises is also increasingly popular as a sustainable alternative to bottled water but you still have to pay for it.
Finally, in Austria, France, and Italy at least, serving a complimentary glass of water with coffee is common and accepted, either as a matter of course or on request. Cafes should therefore be perfectly OK with it.