Why not? Companies that make pharmaceuticals, prosthetics, imaging devices, etc are all on the stock market too, so if hospitals weren’t on there, you could build a portfolio to approximate it by buying producers of medical equipment.
The real issue isn’t whether something is publicly traded, but collusion between groups to keep prices high. For example, it’s mutually beneficial for insurance, hospitals, and medical equipment providers to increase costs. Higher equipment costs means care providers can charge more (what’s another few hundred when the bill is in the thousands?), and higher total bills means insurance companies can charge higher premiums (they’re usually limited to a certain percent of cost as profit). Hospitals generally don’t have direct competitors since it’s prohibitively expensive to build one and there’s lots of bureaucracy based on “need,” so you can’t just go next door to an org that’s not involved in the collusion.
There are lots of viable solutions here, but banning them from the stock market isn’t going to solve anything. The first order of business imo is making everything more transparent.