I would probably always say "for 16.8.2016". When you say "on 16.8.2016", it could start to sound a bit ambiguous as its not clear whether he made the appointment on 16.8.2016 (as in contacted the clinic on that day) or for 16.8.2016 (as in contacted the clinics some other time and asked for an appointment on that date)

"On" sounds more natural here, but when speaking casually I think a lot of English speakers might still say "for" in that sentence. So you could get away with either. "On" would be the proper word to use though.

"On" is for scheduling something on a specific day or date, "for" is more for a purpose. But, it is acceptable to substitute "for" for "on" in this case and it sound quite normal. On is more grammatically correct, though.