2000 IRC 305.1 "Habitable rooms, hallways, corridors, bathrooms, toilet rooms, laundry rooms and basements should have a ceiling height of not less than 7 feet (2134mm). The required height will be measured from the finished floor to the lowest projection from the ceiling". There are three exceptions, but they don't apply here.
The UBC has a higher requirement, but doesn't say if the measurement is from the ceiling or the projection (it also requires 7' 6").
Couldn't find anything in the NEC.
Based on this, I would say that the AHJ has adopted the IRC and he's correct. But, I would ask him to quote chapter and verse because sometimes the AHJ will adopt a building code and make local changes to it.