A generator can damage an air conditioner if the generator is not producing enough voltage or too much voltage or the generator is running rough.
As long as the generator is properly sized for the wattage of the air conditioner and it has enough starting watts then the air conditioner should be just fine on the portable generator.
Electricity comes from large generators that are basically a larger size generator than the portable generator when the electricity comes through the electric lines from the electric utility.
The amount of watts it takes to run an air conditioner on a generator depends on the Air Conditioner as they use different watts.
However for example if you have a 5,000 BTU window air conditioner then it uses around 550 watts so you would need at least a 1,500 watt generator to start the window AC compressor.
If you have an 8,000 BTU window AC then you would need at least a 3,000 watt or higher generator to effectively run the AC as you need more start up power to get the AC compressor going.
The best window AC for a generator would be a 5,000 BTU window AC that uses around 550 watts if you have a smaller generator.
Larger wattage generators can power larger Window AC units and if you have a portable generator that puts out 240 volts then a 240 volt window AC will also work.
As long as your generator has enough wattage then it can safely and easily run any window AC that it has enough power to operate.
Even if you have a camper trailer or traveler trailer roof top AC and have a big enough generator to run it then you can also use the portable generator to run the RV rooftop AC.
A 5000 BTU window AC uses around 550 watts when running.
However the 5000 BTU window AC uses around 900 to 1,000 watts or so when starting the compressor.
It takes more power to start the window AC compressor than it does to keep it running.
So if you use a generator then you need double or triple the amount of watts to ensure you have enough wattage to start the AC compressor.
If running a window AC I would recommend you have at least a 3,000 watt or 5,000 watt generator to allow enough wattage for startup and also for other things you may need the generator for.
It's better to have a larger wattage portable generator than too small.
I have a 7,000 watt portable generator that provides plenty of power and I don't have to worry about not having enough wattage.
It uses near the same amount of gasoline as a smaller portable generator does but provides more power.
I can run at least 3 window air conditioners if needed off my 7,000 watt generator so it provides me plenty of power.