I don't know of any requirement for supporting LV cables in the NEC other than "workman like manner" and "building fabric". If the cable is something like a hybrid style cable, like an AC or MC based fire alarm cable, then the appropriate article could be referenced, otherwise it's nowhere in the NEC.
I would ask the AHJ what section or article he was referencing for your failure, assuming your cabling is supported by building structure.
he may be pulling something out of context. see section 760.130(B)(1), the last sentence says "Where located with in 7 feet of the floor, cables shall be securly fastened in an apporoved manner at intervals of not more than 18 inches." its out of context to just flaty apply it to all installs. the requirement only falls into this specific situation as discussed in the paragraph. we find the same sentence in 760.53 (A)(1). specifically,securing and supporting cables is in 300.4, and 300.11