I had exactly the same problem. The dots show in any aircraft, any location.
My computer is Windows 10, Intel i7, Invidia GTX650.
My version of X-Plane was originally Beta, updated several times, and the last update gave me those black dots in the distance.
So I deleted X-Plane (all of it, everything) and downloaded a total fresh copy.
That fixed the dots, and it seemed to clear up some other issues I couldn't figure out.