If "percent" is just a quantification noun, then it follows that the count-noun "doctors" is head of the NP "20 percent of the doctors". And according to the 'rule' that count nouns take "fewer" and the rest take "less", then "fewer" would seem to be correct.
But it's very easy to mistakenly see "percent""20 percent" as the head of the NP (like a partitive). And that coupled with the fact that "percent" is singular-only (there's no *20 percents) is likely to lead to the wrong conclusion that "percent""20 percent" is a non-count noun phrase as head, and hence "less" must be correct.
Seems a free choice to me.