A person with a bleeding eye should call 911 and receive emergency care for the eye, notes WebMD. Bleeding in the eye is a serious medical condition that requires the eye to be protected until medical care is received. The eye should be covered using a clean compress, but pressure shouldn't be applied.
Follow-up care might be necessary after the emergency care is received. The American Academy of Ophthalmology (aao.org) has a search engine that enables users to search for eye doctors. One option under subspecialities is "Trauma," which would include doctors who can handle ocular emergencies like a bleeding eye. Users can search for doctors nearest their location.