I have a burst vein in my eye, outer half is red, blood pressure is normal?

If it looks: like red blood in the white of your eye it is a broken blood vessel, usually a capillary, called a subconjunctival hemorrhage. If painful- get examined.

PTERYGIUM: Quite benign. A thin membrane growing over the surface of the eyeball and containing tiny capillaries. Minor abrasion can cause a tiny capillary leak. Dramatic blood color will resolve over 2 weeks. No treatment except reassurance.