What you are describing looks like a subconjuntival hemorrhage (that means bleeding below the superficial skin of the eye).
there are two aspects to this -
1. It can normally occur to anybody and will mostly resolve without any consequnece in a months time. There is no need for any medicines for the same.
2. It can be due to
a. high blood pressure
- needs to be checked with a local doctor
b. bleeding tendencies - this can be there if you bleed very easily and for a longer than normal period when you have , say a small cut on your body etc.
Your doctor would have looked at these things and if they are normal, then there is no need to worry.
Hope this helps. Have a speedy recovery