Creating a national health care system would aim to provide a much greater supply of health care to meet total demand. While this would be expected to reduce prices and therefore doctors' income, in reality, the health care system is far more complicated than the simple dictation of supply and demand.
For example, a recent 2011 study by the American Journal of Public Health found that doctor incomes in Canada were already high relative to other professions before nationalized health care. After the institution of the nationalized system, doctors actually enjoyed a significant windfall, with incomes booming for many years during the adjustment period. Eventually things settled down, but in the years since, Canadian doctors have continued to enjoy high and rising incomes, even if they are slightly less than their American counterparts.
This is certainly an encouraging study. However, it is only one study, and the truth is that there are many factors that could affect doctors' incomes in the U.S. in the event of the creation of a nationalized health care system. Incomes could ultimately be less than they would have been otherwise, or perhaps they could even be boosted a little bit. The most likely scenario at the end of the day, would be that doctors' incomes wouldn't undergo a severely erratic change, but of course, that depends on the nature of the new health care system that is put into place.
Source: Duffin, J. (2011): The Impact of Single-Payer Health Care on Physician Income in Canada: 1850-2005. American Journal of Public Health, Vol. 101, No. 7.