I guess your doctor would have said your kid has sub-conjunctival Hemorrhage.
It means the tiny blood vessels in his white part of the eye have broken resulting in small collection of blood.
It can happen due to various reasons like injury/ excessive rubbing of eyes, excess coughing/ vomiting
etc, some eye infections etc.
If no other associated symptoms are seen, you need not worry.
It might take 1 or 2 weeks to resolve.
How-ever if you see new spots coming up/ he develops any other bleeding tendency etc, please do not hesitate to see your doctor again.
Hope my answer helps you.
Wish your son good health.