Well, how would you design a tax scheme to pay for it? Remember, a tax that doesn't directly derive from healthcare would be unjust according to these axioms. So it can't be an income tax, just as it would be unjust to pay for roads using an income tax. You tax everyone using the healthcare system and to the degree that they use it and pay for it that way.

Further, healthcare isn't a commons, so it is arguable that a state has no business interfering. Emergency facilities specifically are a commons, if the hippocratic oath is followed and all in need are helped. So some scheme where people pay for the maintenance of these facilities and services rendered in them based on how much they stand to benefit from them would be just. I don't know how you'd do that beyond sending them a bill. Maybe yearly renew on an access card, like vehicle registration, that isn't based on income but maybe could be based on medical history, or just flat rate for everyone. Taxing income, or wealth, to pay for blanket coverage including person to person contracts with specialists would be unjust, socializing something that isn't a commons would be unjust, but managing emergency facilities as a commons and funding them with some type of tax directly derived from their existence would be, because they are a commons and do suffer from tragedy of the commons.