After the Christian faith became legal in the Rome Empire, Christian hospitals were founded. Christians have always been in the vanguard of medical care.
But there are currently secular forces which, if they have their way, will drive Christians from the medical profession. I mean in the Western world. Christianity is expanding in the Southern hemisphere. Whether they succeed remains to be seen.
There is, however, a possible countervailing trend that lies ahead. Superbugs are on the rise. We have failed to invest R&D in the next generation of vaccines, antivirals, and antibiotics. In the not too distant future it's possible or even probable that civilization as we know it will revert to pre-antibiotic, pre-antiviral conditions. Revert to the time when pandemics killed thousands or tens of thousands at a time.
If that happens, that will have a winnowing effect on healthcare. Traditionally, in nations influenced by Christian morality, it was an unquestioned duty to care for an ailing family member–usually at home. And this was before the advent of antivirals and antibiotics. This meant that when caring for a relative with a contagious, life-threatening disease, the caregiver was a high risk of contracting the same illness. And there was no effective treatment.
It wouldn't surprise me if many family members who cared for sick relatives succumbed to the same illness. They were in constant physical contact with the the sick, infectious relative.
Likewise, Christian physicians, nuns, and nurses used to care for sick strangers. And they jeopardized their own lives in the process.
I don't know the history of heathen cultures in that regard, but it wouldn't surprise me if pagans carted a sick relative off into the woods to die in isolation. To keep them at a safe distance, rather than care for them.
One might object that this was before germ theory, so that people had less reason to fear of infection. But I think that's implausible. Even if people back then had a poor understanding of disease transmission, they could surely observe the correlation. Indeed, even in the OT, you had quarantine laws. People in OT times understood that some diseases were communicable through physical contact.
If the future flips back to the past, who will care for sick people with contagious, life-threatening diseases? How many atheists will put themselves at risk? How many atheists will endanger themselves to care for a sick relative–much less a perfect stranger? Or will the task once again fall to the Christians?