I think this is an interesting question that comes up in this thread from time to time. There are a couple of points I'd like to make.
As the OP mentioned - he incidentally discovered he could not read any of the Ishihara plates (although I hope he could read the control plate..). As many as 9 percent of males fall on the red-green colorblind spectrum (deuteranomaly) as mentioned above and many on this spectrum do not know they even have an issue because in regular day to day life - no one regularly checks Ishihara plates on you and you don't need such fine color differentiation for daily life. Being colorblind in this form does not mean that you cannot see any colors or that you cannot distinguish the color of grass from the color of a stop sign. It just means there are some subtle distinctions (ie Ishihara plate) that you cannot distinguish.
Pathology residency, General Surgery residency, ENT residency, Neurosurgery residency, Internal Medicine residency, Emergency Medicine residency, Pediatrics residency, etc do not request an Eye Exam from applicants and do not check applicants' color vision. US medical schools also do not check color vision on applicants. There are likely a good number of doctors in all of these fields that are red green color blind (9% of males, if an even distribution among doctors) and many of them probably do not even know they fall on the color blind spectrum because no one ever checked Ishihara plates and told them they had a subtle color differentiating issue.
Should all these doctors be removed from practice? Of course not. Clinical medicine and taking care of patients involves much more than just the doctor's own color discrimination ability. The knowledge base and experience of the physician, the history the patient presents with, the other examination findings - all play a part in making an appropriate diagnosis. For example a pediatrician that cannot easily tell when the tympanic membrane is red, can use other clues including that the child has a fever, is uncomfortable, is pulling on his ear, has a history of previous ear infections, to appropriately make a diagnosis and initiate treatment. In all of these fields there are other clues that you can use. Ophthalmology is no different.
I believe the reason this question arises in Ophthalmology more than other just as complex and detailed microsurgical fields such as ENT and Neurosurgery, is that Ophthalmologists are checking Ishihara plates on their patients from time to time and so are more likely to notice if they have an issue. Also a few Ophthalmology residency programs request an Eye Exam from applicants, although the majority (upwards of 95+ percent of programs) do not.
It seems ridiculous to think that being color blind should hold you back from practicing Ophthalmology but shouldn't hold you back from practicing Neurosurgery or Pathology. Ophthalmology is no different.
As far as legally - your residency training program and the American Board of Ophthalmology have the responsibility to determine who is capable of practicing Ophthalmology. If you were incapable of providing quality medical care to patients - it is the job of your residency program and these medical boards to catch that. 95+ percent of residency programs do not think this is even worth checking for. This means that 95+ percent of residency programs will gladly train you. The American Board of Ophthalmology and all State Medical Boards in the country also do not see this as a hindrance to providing quality medical care to patients. These are the experts and these boards have been around hundreds of years and their opinion on who should be practicing matters much more than a jury of 12 laymen who never went to medical school or examined a fundus.
It is definitely a tricky situation because although it is conceivable that a physician in any specialty without a great ability to distinguish certain colors may be at a disadvantage during their exam, the fact remains that there are so many other physical exam findings, history, and other clues involved in appropriately diagnosing and treating patients that do not rely on fine color discrimination. The fact also remains that almost everyone's color vision abilities are on a spectrum (remember the white and gold dress versus the blue and black dress internet image a couple of years ago). The fact also remains that in hundreds of years of medicine in this country, the medical boards and residency programs do not see this as a hindrance to providing quality care.
This should not hold you back from practicing Ophthalmology. I think you can be a fine eye surgeon and physician.
If, however, this issue still bothers you personally despite the facts outlined above, then I would consider applying for something else.