I've come to a conclusion.
In my opinion, if America is to keep a privatized health care system then it should abolish Medicare and health care for the disabled.
However, I think we should adopt a national health care system, in which case we would no longer need Medicaid or Medicare. The expenditures for Medicare (over $300 billion) should be more than enough to cover the overall costs of switching to national health care.
As a side note, I also believe Social Security for the elderly should become privatized. If you don't invest properly for retirement while you're young and working then too bad. The government shouldn't have to make up for it.