Why does one eye slant down and have weak bone projection? (Photo)

Your lower lids are positioned slightly lower than optimal and these can be repositioned with a canthoplasty, additional support with fat injection or malar implants that extend above the orbital rims. A combination of the techniques may afford you the best outcome. You may consult a craniofacial surgeon who specilizes aesthetic surgery.

First off, we all have at least a small degree of facial asymmetry. I would say the amount of asymmetry shown in your photos is very mild.Lower eyelid "retraction" refers to a situation when the lower lids sit in a lower than optimal position. A thorough examination is necessary in order to determine the best way to address this. A canthoplasty can certainly be helpful in some situations. When a patient has prominent eyes (which project forward, which it appears you may have), sometimes it is necessary to place a graft in the lower lid to give vertical support as well as a canthoplasty. Another less invasive option for some patients is filler injection into the lower eyelid, which when placed properly, can also give vertical support and help with retraction.I would suggest having a thorough examination with an experienced oculofacial plastic surgeon to determine your best option. 

Almost all of us have asymmetry between our two eyes so this is common.  Your surgeon is right and a canthoplasty can be performed to elevate the outer portion of the right lower lid and make the  outer corner higher than the inner corner. This is exactly what a canthoplasty is designed for. Good luck.