The Church and the Care of the Mind and Body in Western Culture
Too many people don't realize how old an institution the University is. Oxford, for example, got it's charter in 1214, the year before the Magna Carta was signed. That's why C.S. Lewis could refer to the buildings in which his offices were housed as the "New Buildings" even though some were built in 1650! Perhaps the oldest university--the University of Paris--grew out of a guild of scholars surrounding Notre Dame Cathedral. This is where Saint Thomas Aquinas taught (lived 1225-1274). Most of the earliest universities grew up out of these guilds of scholars around Cathedrals or major churches. They were supported by Popes and Kings and nurtured into the great universities as we have come to know them. The idea caught on.
(No, those aren't churches, but you're close: it's Oxford University and Holy Cross Hospital)
Hospitals developed in much the same way. For obvious reasons (or perhaps not obvious to some readers: I've been a university student, a professor, or both for 16 years now) I'm less apt to summarize the particulars of the hospital so I quote Wikipedia at length:
The Romans created valetudinaria for the care of sick slaves, gladiators and soldiers around 100 BC. The adoption of Christianity as the state religion of the empire drove an expansion of the provision of care, but not just for the sick. The First Council of Nicaea in 325 A.D. urged the Church to provide for the poor, sick, widows and strangers. It ordered the construction of a hospital in every cathedral town. Among the earliest were those built by the physician Saint Sampson in Constantinople and by Basil, bishop of Caesarea. The latter was attached to a monastery and provided lodgings for poor and travelers, as well as treating the sick and infirm. There was a separate section for lepers.
Medieval hospitals in Europe followed a similar pattern. They were religious communities, with care provided by monks and nuns. (An old French term for hospital is hôtel-Dieu, "hostel of God.") Some were attached to monasteries. Others were independent and had their own endowments, usually of property, which provided income for their support. Some were multi-function. Others were founded specifically as leper hospitals, or as refuges for the poor or for pilgrims.
These are two examples of the way the Church nurtured Western Culture. There are many others, of course, (one of my favorites, of course, being the preservation of most Greco-Roman classical literature by Irish monks), but from an institutional standpoint, they are clearly the most visible. Of course, the greatest respect in which the Church has nurtured culture is invisible. We can't expect to receive much credit for that.