Is it safe to use gentamicin ophthalmic solution to cure a stye?

Yes: Gentamycin is safe for eye infections like stye.

Yes, but...: Yes as long as it was prescribed by your eye care professional. Do not use other people's eye drops.

Yes - But: Always try hot compresses and eyelid scrubs with baby shampoo first, often works alone. Antibiotic drops or ointment may also be required but there are better choices that gentamicin alone- a combination antibiotic and steroid such as blephamide or tobradex (tobramycin and dexamethasone) for example, and only on the recommendation of your eye care doctor.