Usually when they "injure" their eyes, it's actually a scratch on the lens of the eye, which will heal itself with no issues except for the possibility of infection, which your vet covered by prescribing the antibiotic...If the "injury" or "trauma" to your bird's eye was anything more than a scratch to the lens, I would imagine that the vet would have told you exactly what it was, because anything more than a scratch on the lens (which is by-far the most common injury to all eyes, including the eyes of people, dogs, cats, reptiles, etc.) typically requires additional treatment, for example a Detached-Retina, which is also a common injury/trauma to the eye, requires a special suturing procedure and a horribly awkward 24-hour period following where they have to be put under light-sedation and kept under it so that they will stay in a certain position for the entire 24-hours so that the Retinal re-attaches itself....
If your bird actually had any type of ***** "discharge" from the eye, then they were already suffering from Conjunctivitis, or bacterial infection of the eye...And if there was a discharge from both eyes, then both eyes had bacterial infection already. A scratch or other trauma to the eye will not result in a ***** or thick discharge unless it's infected already, and this would have been confirmed if BOTH EYES, even the non-injured eye also had a discharge, as bacterial eye infections easily and quickly spread from one eye to the other.
The eyes are probably "watering" because you're still putting the antibiotic ointment in both of them...If you finish the week-long antibiotic treatment in both eyes, and they continue to water, or if the thicker "discharge" comes back and starts again from either one or both eyes, then you need to immediately start the antibiotic again once daily and call your vet back to tell them that as soon as you stopped it the discharge started right back up. When that happens it's usually because the spectrum of the antibiotic wasn't quite right, it was in the correct "family" of antibiotics because it was able to cut-down the infection dramatically, but not able to fully kill it off. If that happens they'll probably want to change antibiotic ointments/drops, which is the correct course of treatment, however if the discharge does come back after you finish the week of treatment with the Gentamycin ointment, do not just allow the vet to prescribe another antibiotic and start giving it to your bird, instead this time you want to demand that the vet actually take a swab/culture from his eye and send it out to a lab (running a "Culture and Sensitivity") so that they can determine exactly what bacteria strain is causing the infection, and exactly what antibiotic that particular bacterial strain is "sensitive" to and will kill it off completely. Don't just go from antibiotic to antibiotic without a Culture and Sensitivity being run, which usually take about 3-4 days after they take the swab and send it out to a lab to get the results.
****Also, I'll say this again since his eyes are still "watery", which again is most-likely due to the antibiotic still being used, but just in-case the discharge does come back after you finish the week-long treatment, please pay close attention to your bird and if you hear any coughing, sneezing, wheezing when he breathes, "clicking" when he breathes, if his breathing seems at all labored like he's having to work to breath and you see his chest/belly/shoulders going up and down heavily, or if he starts to "open-mouth' breathe at all, or if you see any discharge coming from either nostril or any sneezes are wet, then this is when you need to either find a Certified Avian Vet, or you need to take-charge with this Vet that you already have (who seems to know what they're doing so-far so that's good), and demand that they take cultures/swabs and test your bird for Psittacosis Chlamydia, because the 2 main/first symptoms of Psittacosis Chlamydia are #1) Upper-Respiratory symptoms that I just described above, and #2) Conjunctivitis in BOTH EYES. I don't at all think that is what is wrong with your bird, I think he scratched is eye while flying and it got infected and spread to the other eye (if they both had thick discharge coming from them), so don't worry about it being anything more than that. You haven't described any Upper-Respiratory symptoms at all, so he's fine...I just want to make sure that you know that any time a bird (any species) has Conjunctivitis, you want to watch closely throughout the time they have the Conjunctivitis for any Upper-Respiratory symptoms going on, because Psittacosis Chlamydia is a bacterial infection that is very treatable, but it takes a LONG, LONG COURSE of oral or injectable Antibiotics to get rid of it, and it comes right back in about 90% or more of cases because the Vet doesn't do a long enough antibiotic course, or they use the wrong antibiotic (usually it takes a 48-day course of Doxycycline, or Doxycycline combined with Azithromycin, so it's a horrible pain in the ass)...So it's a pain in the ass to treat, and you want to get a jump on it. But I don't think your bird has it at all, just an FYI...