At great cost, yes you could. This is one of the reasons everything costs so much in America, because we insist on making 50 independent systems.

Then why healthcare isn't greatly expensive in Europe, with, you know, 40+ independent countries?

Because you pay your taxes to the central entity that is running those programs. In the US, the majority of taxes go to the Federal gov't, not the states.

The federal government should then stop levying those taxes, so that states could then collect more and run their own programs. The federal Medicare tax is already so high that the federal healthcare spending per capita is already exceeding healthcare spending per capita in almost all European countries, yet Medicare/Medicaid only cover a fraction of the population. Seriously, with the same amount of money (in per capita terms) that US government already spends on healthcare, UK or France are able to provide universal coverage. If the federal government is too inept to do the same, it should give up trying, and instead make space for states to do that instead.

Most European countries are of the size of US states. If they can run their healthcare just fine, it only makes it dumber idea to try to centralize it in US at the federal level.

In Canada, the single payer healthcare system is Provincial, not Federal, and yet the income tax brackets at Canada's Federal level are roughly equal to those in the US (in fact are actually higher for some earners).

If we wanted to take Canada's cue, there's plenty of room to increase State tax rates even if we did nothing about Federal taxation. Or we could take Switzerland's cue and totally flip the distribution, where Cantonal taxes far exceed Federal income tax.