If you are referring to modern hospitals with names like "St. Luke's" or "St. John's", I can answer this by saying that this is not a result of a naming convention or history, but hospitals and medical centers managed by large nonprofits organization directly connected to a specific church.

In the same respect, there are many Jewish Hospitals that are owned and operated religious groups. There is a nice, concise piece about the development of the Jewish hospitals, which will give a sense of why some of these organizations went into the healthcare business:

Halperin, Edward, "The Rise a Fall of the American Jewish Hospital," History of Academic Medicine 87, no. 5 (2012): 610–614

Hospitals with municipal or nonreligious names are often run by nondenominational organizations.

However, if you are interested in why these christian organizations in particular might have become healthcare providers, then we can turn our focus there. I have read multiple times that the First Council of Nicaea in 325 C.E. instructed Bishops to place hospitals in every cathedral across Christendom. However, I have never seen a good citation on this reference, and I have never been able to find this information when looking through the texts from the First Council either.

The Catholic Church's interest in caring for the sick is well established though, and certain Popes have stressed the promotion of healthcare as their main causes. The wikipedia article on this subject provides a wealth of information for those seeking more. The Roman Catholic Church continues to be the largest healthcare provider in the world.

It is also worth noting that hospitals in their current form did not come about until major improvements in sanitation and therapeutics in the late 19th/early 20th centuries. I'm glossing over a lot of nuance and history here, but before those developments, hospitals were more a place to die than a place to get healed. Given that history and considering Christianity's connection to the afterlife, I have always thought religious organizations' connection to hospitals were understandable.

Historically, hospitals have always been associated with the church. During the medieval period, they set up "hospitals" but they are not the same kind of hospital that we think of today. They were mostly for women and children, and were run by women of the church or volunteers. They also housed the poor, pilgrims, etc. and didn't necessarily give them health care, but rather shelter. Because Jesus was a healer, monks and nuns were also given this role. There are a few books on hospitals in Medieval and Early Modern Europe if you want to read more in depth.

Mending Bodies, Saving Souls by Risse.

Florence and Its Hospitals by Tombaccini, Lippi, Lelli, and Rossi.

Origins of the English Hospital by Watson.

These are just a few examples.