Hospitals in ancient Rome were generally limited to military camps and the very late empire, after the establishment of Christianity. While legionary medical facilities were quite extensive, hospitals, as we know them today, simply didn't exist in the Roman world. Romans in general terms regarded disease as an affliction of the gods requiring prayer, sacrifice and pagan rites to alleviate.
Prior to the Christian era, there were temples, such as the Aesculapium on the Tiber Island, where the sick spent the night in prayer to the gods hoping to receive a cure, and 'doctors' made rounds doing what little they could. And there were establishments to house the dying or infirmed, essentially to keep them off the streets. The concept was not cure or even care, but to keep the wretched and sickly poor off the streets. Out of sight out of mind, so to speak.
Prior to the hospital concept, wealthier estates may have had valetudinaria attached to the grounds. This was a sort of medical facility to deal with sick or injured slaves and to isolate them from the rest of the staff and family.
The concept of the modern hospital (the actual care, 'hospitality' and treatment of visitors) for the civilian masses in Europe didn't come to fruition until post Constantine and the rise of Christianity. While these early Christian hospitals were grossly over their heads regarding medical capability (they essentially served as last stops for the dying or quarantine centers), the concept of providing care to the public was the actual intent. In this regard, the first civilian hospitals were developed.
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Augustus Caesar assigned four doctors with a supervising legionary physician to each cohort. In addition, the soldiers themselves were well versed in first aid.