Looking at a list of the largest US hospitals, i noticed that several of them have “Methodist” or “Baptist” in their names. I assume that today they would accept anyone of any religion (maybe obligated?). None on the list were labelled “Catholic”, however.
Are these names kept by tradition, or does religion still play a part?
There are a lot of Catholic Hospitals and they don’t just cater to Catholics.
Many are St. Soandso Hospital. It looks like the Catholic Church owns about 14.5% of all acute care hospitals in the nation.
1 data point: My Jewish MIL gave birth to my Jewish wife in a Catholic Hospital.
There are lots of Catholic hospitals. They will take patients of any religion, but there may be restrictions on the procedures performed, such as abortion or contraceptive surgery.
But religion does play a part. I worked for years for a Catholic organization that ran a medical clinic. No birth control services available-- pills, IUDs, even condoms. Needles to say, no abortions.
Ninja’d.
And a few Jewish ones too
Long Island Jewish Hospital is pretty well known I think.
My GP and my endocrinologist are affiliated with Loyola Medicine, one of the big hospital/clinical networks in the Chicago area. The name doesn’t sound Catholic, unless you are in the know that the organization (and the university with which they are affiliated) is named for St. Ignatius of Loyola, who was a Catholic priest, and one of the founders of the Jesuit order.
And, as @What_Exit notes, a lot of Catholic hospitals are named “St. [Somebody],” rather than explicitly having the word “Catholic” in the name.
Also, there are hospital networks in several regions of the country which go by “Adventist Health” or “AdventHealth” – those are affiliated with the Seventh-Day Adventist Church.
Nitpick, as your cite says, some of those Catholic hospitals are affiliated with the Catholic Church but are not owned by it.
And I wonder if the hospitals with Baptist or Methodist or whatever in their names do so because people might assume they’re Catholic hospitals?
Possibly, i suppose, but, for example, I’m familiar with “Brookwood Baptist Health,” a hospital network in the Birmingham, Alabama area. The deep South (save for parts of Louisiana) isn’t particularly Catholic, but it is pretty heavily Baptist, so I’d be surprised if people in Birmingham would have had any thought that their hospital might have been affiliated with the Catholic Church.
I think it’s more likely that Protestants just don’t have much use for saints.
That could be it, too. A Catholic-affiliated hospital can use the names of saints, religious orders, etc. to signal its affiliation (to those who care about such things), but Protestant-affiliated hospitals seem to be more likely to rely on the name of the denomination.
The one non-Roman Catholic denomination hospital I know of named for a saint names the denomination - St John’s Episcopal. Not sure if that was specifically to avoid confusion with a Roman Catholic St John’s that also existed or more generally to avoid assumptions that St X must be a RC hospital
Hardly. 
Like any other religiously-affiliated organization, they will include the name of their religious affiliation in the name because it is integral to the mission of the hospital and they want the public to know this. I doubt if they give Catholics a thought. I also worked for a large Methodist health care system. Their mission statement includes, “Serving Humanity to Honor God.”
We have a hospital in Chicago that used to be named Swedish Covenant (the name of a particular Protestant denomination) until they recently dropped the “Covenant” and are now just “Swedish Hospital”. They do accept patients of all ethnic backgrounds, of course, and AFAIK the cafeteria does not serve lutefisk.
I know of a Chinese hospital in San Francisco and that Wikipedia article mentions ones in Toronto and Montreal.
Hospitals are for curing patients, not fish.
In the past, running hospitals and care of the sick, especially the chronically sick, was one of the functions of organised religion - there were no state-run hospitals.
Which is why plenty of UK NHS hospitals are St. Somebody’s, from their original foundation, but are entirely secular. Though they will usually have some sort of multi-faith chaplaincy available.
My mother spent her last days in a coma, after several heart attacks, at the local Catholic hospital. As I remember “pulling the plug”, or taking her off the respirator, was not an option. I don’t recall whether or not she was showing any signs of brain activity during this time, so I can’t say whether this was a factor in the policy.
The Jewish General Hospital in Montreal serves eveyone, although much of its funding comes from the local Jewish community. And yes - its cafeteria is strictly Kosher.
I remember seeing a Shabbat elevator at Mount Sinai Hospital in Manhattan.