My left eye was pushed further into the socket from a punch I received. Is there a way I can fix this without surgery? (Photo)

Did you get imaging such as a CT scan after the trauma? You very well may have fractured your orbital floor, which can lead to a condition known as enophthalmos. You should seek out a fully trained plastic surgeon for a formal exam. This can be repaired if so. If not repaired, it can continue to worsen. Some patients even develop double vision.

Try to stay out of fights from now on OK.
perhaps the person who assaulted you is simply highly undesirable bully but this fighting can lead to permanent disfigurement and worse.
Most likely you developed a tripod fracture which is very common when being struck in the cheek.
It’s a fracture that includes the zygoma, maxilla and orbital floor.
If the punch strikes the eyeball itself a blowout fracture can occur without fractures of the zygoma.
Ideally these are treated within the first few days after injury after being evaluated with a CT scan.
The fractures push the bones apart creating an eye socket that has greater volume and can lead to the cheek looking pushed in.
Most of the time the outcome is only Aesthetic and doesn’t cause problems with vision or other problems.
It’s much more complicated to correct these injuries after the bones of healed.
Sometimes a different approach is taken that simply focuses on decreasing the volume inside your eyesocket which pushes your eyeball forward.
The first step is to seek the consultation of a board-certified plastic surgeon who regularly does facial trauma surgery. You may find these at hospitals that serve as a level one trauma centers or academic medical institutions that have medical schools.
they will most likely order a CT scan or possibly an MRI.
After that elective surgery can be scheduled.
For future reference please seek medical assistance sooner rather than later after facial injuries.
A classic sign of a tripod fracture includes scleral erythema(The white part of the eye turns blood shot in the outer half of the eyeball.)
At this point there’s no hurry to get this fixed so take your time and meet with a few different plastic surgeons. Medical insurance will most likely cover the cost of this to so look into your coverage and what it includes or doesn’t include.
Best, 
Mats Hagstrom MD