I've had brown, popped blood vessel in my eye for a couple of years. can it be removed?

Probably not: Brown spots on the eye that do not disappear are usually not blood but rather melanin - a brown skin pigment. The white of the eye is covered by a thin layer of skin that can have melanin (which makes the skin brown), more often in darker-skinned individuals. Have it evaluated annually to be sure it does not change. It is only removed if there is concern about growth.

Sounds like a nevus: Or mole in the eye. If it has changed it should be evaluated. Blood vessels are not brown.