St. Fabiola was a 5th-century Christian Roman noblewoman whose dream was to open a hospital in Jerusalem. She had to settle for Rome instead, creating Europe's first hospital.

A mural at the site of Europe's oldest hospital building

Santa Maria della Scala was founded in Siena, possibly a thousand years ago. It largely served pilgrims' health needs but was also a major repository for abandoned babies.
One of the earliest still-functioning hospital foundations in the world is the Hotel-Dieu in Paris. Founded more than a thousand years ago, it is now a state-run institution, although it was staffed by Augustinian nuns up to the 20th century.

Another Hotel-Dieu hospital

This one is Canada's oldest, founded in 1645 when Montreal was first colonized by the French. It is still running today, although not as a religious institution.

St. Bartholomew's Hospital in London

Henry VIII destroyed much of the Catholic healthcare system in England. St. Bartholomew's Hospital in London was exempted and remains one of the UK's premier hospitals, although "St. Bart's" is entirely secular these days.
The oldest continuously used hospital in North America is probably the Jesús Nazareno of Mexico City. It was opened by the Spanish conquistadors for the treatment of wounded Aztec warriors.
The original Mayo Clinic in Rochester was founded as a collaboration between an Englishman, William Mayo, and the local Catholic Sisters of Saint Francis of Rochester, Minnesota. It started as Saint Mary's Hospital, as seen in this photo from 1910, before becoming the world-famous Mayo Clinic.