I would not be concerned is you are not having any other problems such as blurred vision, pain, or discharge. A subconjunctival heme may increase in size then resolve in about a week to 10 days.
Dr. O.
The problem I mentioned is called "Sub-Conjunctival Hemorrhage"
The most likely cause of a red spot on the white part of your eye is either:
1) A small amount of blood from a broken blood vessel.
or
2) A focal area of dilated vessels.
The white part of your eye is called the sclera, and it is the shell of the eye. There is a thin membrane above it, which is mostly transparent that is call the conjunctiva. That layer contains the blood vessels. Sometimes at night we rub our eyes without knowing and break one of the fine delicate vessels in the conjunctiva. A small amount of blood collects between the conjunctiva and the sclera.
This small spot of blood can move down with gravity, and may even increase in size. They are essentially harmless. Taking aspirin, ibuprofen or mega-doses of omega-3 can make you more likely to have small bleeds. If you take warfarin, you might consider getting your INR checked to ensure it is not too high.
It never hurts to get checked, but I wouldn't rush to the emergency for this problem.