Is the American osteopathic board of ophthalmology an official certification body for hospital privileges?

The full answer to these questions will take me a while to explain. The short answer is yes to your first question and no to your second question. Now for the rest of the story:

There are two parallel types of physicians in the US: MD (allopathic) physicians and DO (osteopathic) physicians. Both paths require essentially similar pre-med requirements and both require four years of medical school. The DO medical schools require additional education in musculo-skeletal relationships and their impact on visceral disease. Both tracks are followed by an internship and residency training. MD-track physicians who meet the requirements and testing may be certified by the ABMS (American Board of Medical Specialties). DO-track physicians who meet similar requirements may be certified by the AOA (American Osteopathic Association) upon the recommendation of their specialty Board of Examiners. The American Osteopathic Board of Ophthalmology is such a specialty Board of Examiners. The ABOO does not actually award the certification. That is done by the AOA. DO physicians who are certified by the AOA as above, are recognized as board-certified specialists in their field of practice in all aspects of medicine and surgery no more or no less so than MD–ABMS certified physicians. They are equally regarded by the US government, Medicare, the military, all hospitals receiving Federal funds, etc. Discrimination based on DO- or MD- track can and has led to serious litigation.

As for the second question: the ABMS does not certify DOs who complete DO residency training programs. There was a period of time when DOs who completed DO training programs were allowed to take the ABMS written and oral certification exams and were certified by the ABMS. This practice was terminated by the ABMS around 1980 at which time foreign medical school graduates and DOs were no longer allowed to take the examinations. There are DO specialists from the era before the prohibition that are certified by both the AOA and by the ABMS. Since that time, however, DO-trained physicians have not been allowed to take these examinations, at least with regard to most specialties of Medicine. Some DOs have completed MD residency programs and have gone on to be ABMS certified in their specialty.

Now, just to make this answer more complete (and perhaps more confusing), it appears that in the very near future all training programs (DO and MD) will soon come under the control and management of ABMS. So, instead of two parallel tracks, there will be two tracks (DO and MD) that merge into one ABMS track insofar as specialty training and certification is concerned.