Glad it all worked out well in the end!
Just for future reference, you might find this map useful.
Here you can find all the (public) primary healthcare centers in Galicia. In any of those, there should be both doctors and nurses able and willing to help in these circumstances. Queues should not be so long there as one would expect in hospitals, which are more scattered in the territory and typically devoted to more specialised services within the Spanish health care system. If a private insurance (or EU card) works in a hospital, my guess is that it should definitely work in the primary clinic too, since they both belong to the same system. Should you require more specialised attention, no doubt hospitals are the first place to go. You can also find them in the previous link.
Similarly, other regions in Spain will have their own web of health centers (competencies in health issues are transferred to regions,
Comunidades Autónomas, in Spain). They come under names such as
Centro de Salud, Centro de Atención Primaria,
Centro de Saúde, Centre d'Atenció Primària, or the like.
In any case, I wish you do not have to use much of these services as it will mean that you are having a smoother
camino!