not with a state doctor, as far as I know - not atm anyway - so yes, private or urgencias, though you'd likely be charged there, too

there was talk of a nationwide contributory scheme, like we used to have in Valencia, in the last round of changes, but that doesn't seem to have happened yet afaik

Tony & Janis, you need to try and find a bilingual person locally who will help you through the paperwork till your Spanish is better. Try a notice in the town hall or in the local secondary school (IES), there are bound to be students desperate to earn a few extra euros. The going rate round here is €10 for half an hour's interpreting.

The EHIC is only for urgent and necessary treatment - not for routine medical care. Treatment is given so you can return to your home country for treatment. So, as a tourist or temporary visitor, not if you are settling permanently
If you are making a permanent move, then you would need your S1 form, from DWP, assuming you have paid into the UK system whilst you have been working overseas. The British Embassy gives quite clear guidelines for this on their website