Is a horse scared when you see the whites of his eyes?

Often. The way a horse's eye is structured, you don't normally see the whites at all - unlike human eyes. The only time the whites are visible is when the horse is scared or over-excited and their gaze is moving all over. If you look very closely you can see a little bit of white.

There are exceptions. If the horse has less pigmentation around the eye, then the white of the eye also becomes more visible. This is most common in Appaloosas and horses with similar coloring.