Can this eyeball asymmetry be corrected? (Photo)

To fix this would require surgery to elevate the globe.  This would be either a surgery on the bony orbit, or some sort of an implant under the eye to lift it up.  There are major complications involved with these surgeries, so you would have to think long and hard about doing this just to become more symmetrical.  See an oculoplastic surgeon to discuss your options. 

You need a full oculoplastic specialist for evaluation. The left eyeball seems more down than the right side (hypoglobus). That could relate to an orbit problem. 

This assumes that your doctors have fully worked you up before treating you cosmetically. Correcting this requires craniofacial surgery.  It is essential to understand that craniofacial surgery is not cosmetic surgery.  In my opinion, craniofacial surgery is not worth the risk associated with this surgery for your issues.  Typically we use crainiofacial surgery to make human beings whose faces are not human appearing look more acceptable.  While you can see your asymmetry I suspect that very few people who you interact with are aware of this real but subtle difference.  It is also possible that you have body dysmorphic disorder and this real facial asymmetry the focus of your concern.  This is a psychological condition where you spend excessive time concerned about a physical issue such as thinking about it for many hours a day, avoiding social situations due to concerns about how other perceive you.  Working with a psychologist can be very beneficial.  It does not change your appearance but it help your sight into your feelings about it and that makes a huge difference for people.

You have significant orbital dystopia that is congenital.The first place to start is to get a 3D CT scan to have a clear idea as to the differences in the orbital bony asymmetry. This will determine how much differences exist in the level of the orbital floors and the size of the 'orbital box' on both sides. This will determine how much improvement can be obtained by orbital floor augmentation and whether it would be worth the effort. Despite your degree of orbital dystopia it is not worthy of more major orbital box repositioning through a craniofacial approach.