This diagram troubles me. At the left we see Ground/Neutral coming in. There is no ground/neutral. It will be either ground (earth/grounding conductor) or neutral (grounded conductor). The only place these are joined (and allowed to be joined) is at the service entrance. The ground wire is there to ground equipment, IOW to be sure that the frames of all equipment connected to the system are at the same potential - earth potential. That is its sole purpose unless there is a frame to hot or frame to neutral fault in which case it serves as a return path for the fault current as opposed to a path through someone who touches the frame. Sticking with the case where the 3rd wire is ground/earth one can 'convert' to 4 wire by installing a center tapped transformer in the 'box'. The neutral connects to the center tap. One may not 'derive' a neutral by connecting a white to the ground wire. To do so violates one of the fundamental tenets of the code: the ground wire is not to conduct load current - only fault current. If the loads on the 120V circuits are equally balanced (as is a single 240 heater element) then the loads on the red and black legs are perfectly balanced and there is no current in the ground wire. This is why true 240 only loads are wired with red, black and green. If you connect any 110V load between a hot (Red, Black) and the ground wire the return current from that load will flow back to the panel through the ground wire thus violating the code. Saying that you are not violating the code because you are doing this with something that plugs in and consequently does not, therefore, change the house wiring is fooling yourself. It certainly violates the spirit of the code if not the letter. No 'appliance' that took 120V from a hot, hot, ground branch would get UL approval.
Now lets suppose the three wire circuit is hot, hot, neutral. You certainly can take 120V from that arrangement but what about the ground/earth connection? It isn't available. If you connect the neutral (white) to the frame of an 'appliance' which is one of the few excepted from the requirement to have a separate ground by the code (dryers, ranges and ovens) then you are grandfathered in if the house was wired H/H/N before a certain date. Hooking a green wire to the white wire in a spa panel is not covered by the code and does not 'convert' the system to 4 wire and you are fooling yourself if you think metallic equipment connected to that green wire is properly grounded. It isn't. It may be better grounded than if it were allowed to float but it will be at other than ground potential if there is any asymmetric load (including a fault). A few mV are not a big deal with a clothes drier but when conductive liquids are around it's a different matter. Note that hot to pseudo ground leakage should cause the GFCI to trip if the connection to the neutral is made before the GFCI but not if made after. Also note that in the case of asymmetric load with connection after the GFCI the G-N circuit will trip the breaker if the pseudo-ground has a path to true ground (the N-G detection current loop is from the injection point on the neutral at the GFCI to the neutral P-G connection, through the leakage path to earth through the earth to the service entrance N-G bond and back down the neutral to the injection point). The voltage induced by an asymmetric load (no fault) would not trip the breaker
One would have to run a separate earth wire from the panel or sink another ground rod somewhere near the spa panel (and I'm not sure what the code would have to say about either of those) in order to get a true 4 wire system.
I've certainly not researched spa panels extensively but all the ones I have looked at expect a 4-wire connection to the panel. The reasons for this seem pretty clear to me. Can you get something that 'works' with a three wire connection to a spa panel based on the asymmetrical parts of the load being small compared to the symmetrical? Yes. Is it safe? Not so sure. Is it kosher? Probably not.
