In The New Testament believers are instructed to care for the sick and the poor. And there is no qualification that it applies only to fellow believers. We see this in Acts 5:15-16.
There is a reason many Hospitals are named the same way Catholic Churches are. Because in medieval Times the Churches were the Hospitals. The Hospital the Knights Hospilitar were named after was a Hospital in Jerusalem that was also a Church, all those Crusader orders were linked to specific Churches in their founding.
Now in the first Century we were not yet numerous or wealthy enough to succeed in caring for all the sick and injured of the entire world.
But today the Roman Catholic Church is wealthy enough to provide universal Healthcare to every Nation where they are the majority and/or the state religion. Plus help a great deal in caring for the sick and poor of the U.S. and Japan where they have a significant presence. It'd be nice if the modern descendants of the Knights Hospiliatars, the Knights of Malta, spent less money funding Islamaphobia and Neo-Con foreign polices and more money funding Hospitals.
The Anglican Church can likewise afford to do the same for every Common Wealth Nation, plus help in the U.S. where many American Christians still favor Anglican style Protestantism.
But even without the help of Rome or London, the Mega Churches and largest home grown denomination of the U.S. can afford to provide Universal Healthcare to the entire Nation if they worked to-gather to do so.
Now many Churches are making an effort to still follow this mandate. But the wealthiest ones aren't doing nearly enough. Some Hospitals are still technically run by Catholics, I think it's shameful that any Catholic Hospital has to charge money to provide healthcare.
Since I'm an English Speaker who was raised Catholic, I'm going to limit my comments on the moral responsibility of well known churches to just the Catholic and English Speaking denominations.
This is perhaps also a good time for me to address those sects that teach seeking secular Medical help is somehow a lack of Faith in God. That you should Pray for a miraculous healing and nothing else. By misusing passages like James 5:14-15.
One of the Authors of the New Testament was a Physician, in fact it's said besides maybe Paul, Luke wrote more of it then any other Author. And no this was not his job only before being Saved, he was Saved already when Paul met him at Troy. He served as Paul's personal Physician.
In 1 Timothy 5:23 Paul gives Medical advice, alluding to the Health benefits that Wine can sometimes have.
The Bible even acknowledges the existence of Mental Illness, contrary to popular assumption. Matthew 4:24 clearly refers to the Demon possessed and " those which were lunatic" as separate, not the same thing. But demons may sometimes take advantage of mental illnesses hence Luke 17:15-18. Today Lunatic is not the politically correct term, but it was a proper medical term back then.
The strongest Biblical argument advocates of this dangerous doctrine have is the KJV rendering of 1 Chronicles 16:12.
"And Asa in the thirty and ninth year of his reign was diseased in his feet, until his disease was exceeding great: yet in his disease he sought not to the LORD, but to the physicians."Which is clearly a negative tone.
The word translated "physicians" here should more literally be translated healers. And given the words this word is related to (like Rapha and Raphaim) I think it's clear it sometimes refers to occult healers. In fact I think this situation is meant to parallel Ahaziah in 2 Kings 1 seeking after Baalzebub when he was sick. The condemnation here is turning to a god other then Yahuah.
Going back to the initial topic. My view is The Church should be providing Free Healthcare itself, if The State has to do it then The Church is failing.