Hospitals aren't going to turn away patients while they have open beds, period. Triage isn't about making moral judgments to favor future patients who may or may not be more deserving than someone in the ICU right now.
Regardless, it wouldn't work. The moment hospitals made being vaccinated a requirement for receiving care (which is horrific and would never happen), every single unvaccinated COVID patient would simply lie and say they were vaccinated. Hospitals aren't going to send critical patients home to come back with their vaccination cards before receiving care. The more you incentivize patients to lie in order to receive treatment, the worse things get.