Did you have to pay for the whole visit out of pocket, or just an insurance copay?
I would call the office like the others said and explain, but I wouldn't get angry if they don't refund you. Medicine is not an exact science. Your obviously had "moms" intuition to bring him to the doctors, but my guess is his eye didn't present as an infection at the doctors office, and then it got worse. There is so much overuse of antibiotics nowadays, that some doctors shy away from prescribing without enough symptoms (or a + culture). (and rightly so, since many parents demand antibiotics for their kids for every sniffle).
Tips: Does your doctor have privledges at the hospital you took your daughter to? That may help in your argument, you took her to the next best place when the doctor was no longer available after hours (for an ongoing problem). (a place where the doctor also works).
Don't bother with the office staff, just ask for the billing dept. If you get no where with them (and esp if you did pay out of pocket, and don't have insurance), then I would speak directly to the doctor. Still be as nice as you can, and just explain what you told us.
Good luck, let us know if you get anywhere.
(this post brought to you from a nurse of 18 years!

)