to answer the OP question
No.. as along as the generator is big enough (enough KW), it is going to maintain voltage and NOT hurt the A/C
if the generator is too small, either the A/C will stall or the generator will overload/trip breaker/stall engine
the MOST often occurrence of A/C damage is extensive runtime with low voltage shore power, something that does NOT "naturally happen" when using a generator
a generator under so much load it has "low voltage" is going to "fail" in some manner, as noted above
