You could. From what I can tell online it is more for bacterial infections that are on the skin, respiratory system, and urinary tract. The FDA did also pull the use of this medication for use in poultry via water administration due to the increase of fluoroquinolone resistance in a specific bacteria that is a human pathogen. Using it once should not be an issue but think carefully. I personally would not just since the applications of it does not meet your desired use for it.
There is a specific eye ointment called "Terramycin" that is good for eye infections. This is what I would use instead if you have it on hand. It does require a prescription but it is something that could help with the infection. Unsure what the impact with the vision would be. I've used it myself on a rooster of mine who had an eye issue and it helped.
Disclaimer, I'm not a vet but this is just my two cents. If I were in your shoes, this would be my course of action.