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No. Gentamicin is an aminoglycoside antibiotic, and is not chemically related to sulfa drugs. Don't let the Sulfate suffix frighten you. The manufacturers make the gentamicin as a sulfate salt to increase its solubility.
Sulfa drugs are formally known as sulfonamides, with one widely used product being sulfamethoxazole. Sulfonamides are not as popular as they once were, as better antibiotics are available with fewer side effects.
Using the gentamcin topically in the eye means a very small amount is absorbed into your system. If your daughter was allergic, her eyes would be inflamed.
Without seeing the rash, it is difficult to name the cause. If your daughter does not improve overnight, or if the condition worsens, an ER trip may be in order/
Good luck in caring for your daughter.
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