I don't know how it works in Spain, but in France where illegal immigrants can also get some form of coverage, the process is definitely not done in hospitals:

I don't know how it works in Spain, but in France where illegal immigrants can also get some form of coverage, the process is definitely not done in hospitals: you have to apply for it, and then you are issued a specific insurance card. As far as I know you can't get it if you've been in France for less than three months: it's for immigrants, not tourists.