You run no risk of damage unless the generator malfunctions or is overloaded (and the voltage sags). A non-inverter generator outputs a (theoretically) pure sine wave, in as much as the waveform is generated by rotating a coil through a magnetic field. Distortion and minor noise from e.g. brushes can creep in to some extent, but it's still fairly clean. Were the engine governor to fail and/or the voltage regulator fail in the governor you may get too high an output voltage that could be problematic for some devices (though not a lot of modern electronics which have power supplies that are rated for 100-240V operation), but then again any malfunctioning generator could cause trouble.
The fridge electronics of a standard RV fridge would hardly care what the generator waveform looks like since they run on 12V power. The only 120V part is the heating element, and it's just a resistor and would work just as well on pretty much any 120V RMS waveform, including 120V DC. A residential fridge may be somewhat more picky about its power.