Hi Juliette, as you know you can go to a private pay clinic but I would try going to the emergency room of the nearest hospital first.
When I was leading groups of students here in Spain and had emergencies, I took them to the nearest hospital and was never asked for any type of payment, EVER. That was in Granada about 6 yrs. ago so no guarantees it will work but can't hurt to try.
It also helps if your Spanish skills are minimal
I can recommend "Unidad M�dica" as a private clinic where they are multilingual. It was not cheap nor overly expensive (by US standards anyway

) and the med. attention was good.
Here is the info...
UNIDAD MEDICA
C/. Conde de Aranda, 1, 1st floor.
Metro: "Retiro"
MAP HERE Normal Hours:
Mon. to Fri.: 9am-8pm.
Sat. 10am-1pm.
This being August, their hours change:
Mon.-Fri.: 10am-4pm.
tel. 91 435 1823
fax: 91 577 9499
You should call first to make an appointment.
I am sure since you speak Spanish you could find a cheaper private clinic but I have no personal experience with any GPs (only dentists).
Saludos,
jer...