Googling suggests that DNA evidence shows that about 5% of "white" Americans carry Native American genes (this varies a lot with the region -- for instance, "white" people from Louisiana are more likely to carry Native American (8%) and African American (12%) genes than in some other areas).

200 million "white" Americans x 0.05 = roughly 10 million Americans with some type of NA ancestry. Give that the Cherokee were (and are) quite a large tribe, 700,000-800,000 living descendants doesn't seem at all unreasonable.

Edit: corrected to account for only the "white" population rather than the whole population.