I went to Hospital Britanico on Marcelo T Alvear (between 9 de julio and Calloa). It cost me A$R80 I think and she was a real doctor. Then I went to Pfortners for my lenses and they checked my eyes again. Pfortner's is expensive, but they are thorough.
This is the probably the most respected eye clinic in the city:
Malbran (oftalmologia) located at Parara 164 in Recoleta.
I have used it quite a few times. It was recommended by a leading eye surgeon in the USA (when I say recommended I mean that the clinic was included on a list of the best clinics available around the world).