No, it's the other way around. 'Hospital' has been in use since the 1300s, though it had various meanings usually related to charitable accommodation of some sort to start with. By the middle of the 15th century, it had taken on its modern meaning of a place of healing.
'Spital', originally spelt 'spittle' - though it had nothing to do with spitting! - did not come into use until about two hundred years later and the district name, 'Spitalfields', not until some two centuries later still.