Is it safe to swim with popped eye blood vessel?

Yes. A subconjunctival hemorrhage doesn't limit activity--just have to rule out any secondary causes, and can wear goggles when swimming.
Maybe. If the hemorrhage isn't due to trauma and you are found to be otherwise healthy then yes, it's ok to swim with it. Be sure to use lots of artificial tears to limit the eye irritation.

Related Questions

Why do I get sharp pain on the side of my head, feels like a blood vessels popping, also feeling very weak. A After the pain leaves it feels cold at area?

Neuralgia. An irritable superficial scalp nerve may be the cause of this symptom. This is more of a nuisance disorder which usually resolves without any specific treatment. Less likely would be an inflamed temporal artery. A neurologist is best qualified to evaluate and advise you if this symptom persists. Read more...