Short answer: Nope.
Long answer: Kinda sorta not really maybe. Your DNA matches will based on (more or less) the past 10 generations. So, you won't be able to point to a segment and say, "Yep, that's a William the Conqueror bit right there!"
The ethnicity pie chart goes back "hundreds, maybe thousands" of years. However, the pie chart is very much in beta and hasn't undergone the promised revision. If you use GEDmatch's admixture tools instead, you may see European populations, and eventually you may link those Euro segments to British ancestors who are responsible for your medieval (and beyond) lines, but again, there's no way to say "Oh, look at that smidge of DNA from the house of d'Anjou on chromosome 18." It could come from thousands and thousands of different people.
One thing I can say is that if you have lots of colonial ancestry (as people who find royal roots usually do), you'll probably be very pleased with Ancestry's autosomal DNA test because you'll have a ton of matches. That means you'll have lots of people to compare yourself to as you work out where the DNA segments come from, which is great for finding those less-famous ancestors. (Assuming that you also upload your Ancestry data to GEDmatch and convince your matches to do the same, but that's another story!)