Hi Stuart,
The nearest hospital is in Santiago de La Ribera, and to be attended there you will need your E111 or your private insurance certificate and your passport. They don't speak English, so you will also need an interpreter.
All the taxi driver know where it is, however, if you go there yourself, take the road running parallel to the airport and at the roundabout (with the big block of susspended granit), go straight over. At the second set of trafic lights, turn right and the hospital is on the LHS at the beach front. Entrance to A&E is at the rear of the hospital.
If you only require minor assistance, most of the farmacia's can help out.
There is a "pay on assistance" medical centre in the middle of Los Alcazares, on the LHS direction of the town hall, opposite the farmacia on the same side of the road (about 500m past La Sola garage. They do speak some English.
Best regards,
Paul