This depends to some extent on what "races" there are in your opinion. However, it seems clear that the dominant race in the United States today is still the white race.
Whites dominate the United States in many ways. Though their share of the population is falling, they are still the largest racial group by far. White Americans dominate in the highest political offices. The racial makeup of Congress is "whiter" than the population of the country as a whole. 45 of America's 50 governors are white. Whites are also richer and better educated than Hispanics or African Americans.
Asians present a problem with this picture to some degree. Asians have a higher median household income than whites do. But it is hard to say that Asians dominate the country given that their numbers are relatively low and they are not represented in government to any great degree.