With the given tub load and 50 psf deck load, I've checked the reactions of the 2x10 joist, and 2x12 beam, assuming there is one beam on each side of the tub. You shall add at least 10% to the design forces indicated in the graph below to check the adequacy of the members.
For foundation, you shall divide the post design load by the footing area to get the bearing pressure, then compare it with the allowable soil pressure. Note you shall leave a safe margin for the weight of the concrete footing, which wasn't included in the calculation.
Under tub joist load = (Tub Weight/Tub Area) x joist spacing = (4800#/7'x7') x 8" = 65.3 plf, then add joist and misc. weight rounding up to 80 plf. (*Please double-check the full weight of the tub, it seems low).
Suggest adding bracing between the posts in each direction to resist the lateral loads (wind and from water splashing).
The most critical bending moment of the joist and beam is when the tub is loaded full, the surrounding deck is free of live load (empty).
This is an engineering matter that shouldn't be expected to get a simple answer "Yes" or "No" from a stranger you don't know. Please consult a structural engineer before proceeding. Good luck.