Because the real problem of health care accessibility is that it costs too much, the first step is to reduce costs. To do this, I would suggest the elimination of Section 106 of the Internal Revenue Code that provides a tax exclusion for employer-provided health care. This is because this policy is the main driver that has led to the increased prevalence of insurance as well as increased inclusiveness in these policies. Without the tax incentive for employer provided health insurance, the health care market is expected to reallocate its resources to a more consumer driven system over time, putting patients back in the role of a cost-conscious consumer. Patients paying higher out-of-pocket costs have been shown to reduce unnecessary treatments and pressure providers to reduce costs.
In addition, I would agree with MG that tort reform is needed. At present, I would suggest starting with simply adopting a "loser-pays" system where the loser is responsible for the opposing party's court costs and attorney fees. This will put some skin in the game for potential plaintiffs and likely cut down on frivolous claims.
Finallly, because the transition from consumer driven care may take some time to drive costs down to a level where they would have been in a consumer driven market, proponents of ensuring access to all may argue that these two suggestions do to little. Therefore, I would suggest merging Medicare and Medicaid making it 100% income based (with more liberal "wealth" requirements for seniors over 70). Within that framework, I would modify the system to where it is no longer akin to insurance. In its place, I would leave a system that only covers catastrophic care. For the remainder of needed care, I would advocate distributing health care vouchers that more closely resemble food stamps. These vouchers would be just like cash except they could only be used for preventative care and normal everyday medical bills. The benefit of this system over the current system is that it requires people to self-ration their care because the vouchers are not limitless compared that the current system that facilitates the over-consumption of care.