Can damage to the obturator nerve during total laparoscopic hysterectomy cause the adductor minimus to have complete atrophy to nothing?

Perhaps but why?: Any muscle that loses its innervation may atrophy, but in this case there is collateral innervation from a branch of the sciatic (tibial) & possibly an accessory obturator nerve. Bigger question is function. With physical therapy adduction is usually recovered, especially since there is also the adductor magnus & if damage was not total interruption it will also heal. If concern see neurosurgery.