Possible to Push Eyeball Back During a Blepharoplasty?

Pushing on an eyeball will not cause it to be asymmetric.

However, surgery on the lower eyelid can either bring out an existing asymmetry (the more forward set eyeball will appear even more forward set if the same procedure on the lower eyelid is done on each side without compensating for the existing asymmetry.

It is possible to camouflage orbital dystopia of this type by selective cheeklifts.

The eyelid position should not move after routine lower lid blepharoplasty.  If too much fat was removed, it is possible for that eye to move backwards.  More likely, asymmetery seen after lower lid blepharoplasty can be the result of :

Preexisting asymmetry not detected by patient (look at before pictures)

Asymmetric removal of fat or skin

Photographs would be very helpful to understand what you are now experiencing after blepharoplasty.  It sounds like you are telling us that after eyelid surgery, your left eyelid is more ptotic or droopy after eyelid surgery.  This is a relatively common problem after eyelid surgery.  Your current eyelid surgeon may or may not be able to help address this concern.  However, one thing is for sure.  Your surgeon will be highly motivated to make you happy. I would recommend bringing your concern to the attention of your surgeon and getting their recommendations.  If necessary, consider a second opinion.

You are noticing asymmetry after Blepharoplasty which is not uncommon.  Some of the asymmetries can cause your eyes to look like one eye is more sunken or protruding then the other.  Alternatively, you may actually have one eye more sunken or protruding. Your surgeon should be able to address these issues.  If you are seeking another opinion, be ready to provide pre-operative photos and undergo a full examination. 

During routine blepharoplasty, it is not possible to change the position of the eyeball in the front-to-back dimension. If you are noticing this, it is either because the change in lid shape or size makes it look like that, or you had a pre-existing difference that you did not notice. Reviewing these issues with your surgeon should help.