Maybe...And Probably.
Check the welder's manual for a chart that references input and output currents. The figures you need are I(max) (this is max inrush current) and I(eff) (running amps). Multiply those two figures by the voltage (120 or 240?) and you will have max and running Watts. If the welder needs more watts than the generator can give you will damage the welder and the generator.
Another important note that I ran into myself is "dirty power". Most of the newer "legacy" brand inverter machines have "auto line sensing" which means it will detect the input voltage and compensate automatically given that there is enough wattage. Many economy machines however, do not have this feature and require "clean power". The difference between "clean" and "dirty" comes down to something called "Total Harmonic Distortion" or "THD". Cheaper generators will have "dirty power" and more expensive generators may produce "clean power". Both the welder and generator manuals should have this figure somewhere. Clean power has less than %5 THD. It will only damage the welder if you run it on a "dirty" generator...Unless of course it specifically states in the manual that it can hang.
I have a Miller and an ESAB that do just fine on my Harbor Freight generator. I have an Amazon plasma cutter that currently lives in the scrap heap because it doesn't.
Hope this helps!