In a perfect world, with perfect electronics, there would be no need for a grounding conductor. It's there to deal with imperfections. The most likely case for a ground fault in this installation would be if the black conductor lost some insulation and shorted itself against the frame of the fan. In such a situation, the casing could become electrified, and a grounded person, touching the fan body could receive a shock.
If it were properly grounded, you'd get a short circuit to the ground wire, and the breaker would trip.
However, the fan is mounted to the ceiling, and the possibility of someone touching it is low. (Unlike, for example, the frame of a fridge or stove, where human contact is part of normal operation.)
So, Is it safe? Quite likely. For added safety, turn the wall switch off when changing bulbs. Is it to code? No. -- Unless, as others have suggested in comments, the mounting box is grounded on the other side.
You can however, locate the breaker, and replace it with a Ground Fault Interrupt breaker, which would trip should a ground fault occur. This isn't something you should do yourself unless you have experience working in the panel.
Another possibility, if your switch happens to be chained off of an outlet, which is quite possible, depending on the routing of the wiring on that circuit, is to replace the outlet with a GFCI outlet, and wire the switch to the load side of it.
You CANNOT re-purpose a green wire smaller than #4 as a neutral, even if it is insulated. Is the #6 ground green all the way?
If so running a new neutral in this case is your only option.
The ground wire in the outside portion of the wiring for a hot tub in conduit CANNOT be bare. It MUST be green insulated.
I would use a GFI in the main panel and a regular breaker in the spa disconnect. Easy-peasy.
Best Answer
Use a plastic NM clamp
My solution to this problem would simply be to use a plastic (non-metallic, IOW) NM clamp, such as an Arlington NM74, NM75, or NM840-844, instead of the metal one, since this box is indoors.