Actually, there is no way to directly convert a stepped wave to a pure sine wave. The voltage must be rectified to DC and then re-inverted at frequency (60 Hz). A UPS uses a battery for it's supply source anyway, so it is always a DC source that is inverted. This can be done with transistors and IC controllers, which will yield a step wave, or it can be done with an oscillator circuit PWM circuits, which will yield a pure sine wave. The latter is more expensive.
On the PSU side, all voltages to the computer box itself are rectified DC, so there is no danger of damaging anything there. However, the internal PSU monitoring and control circuitry may or may not be upset by a step wave input. Such circuits can be triggered/switched by looking for a certain slope of the AC signal- others may look for a zero voltage crossing. A stepped wave pattern has many slope changes in a single AC cycle, which would cause those circuits looking at slope to mis-trigger, probably causing high-frequency noise and likely causing the PSU to shut itself down. This all depends on the PSUs monitoring and protection circuitry, so it will vary from PSU to PSU, even for a given manufacturer. I have not been able to readily find information on manufacturer websites that give a clue. To further complicate the question, there are differences among stepped wave circuits, some of which are filtered enough that they will work fine with some PSUs but perhaps not all.
The bottom line is you won't know until you test your exact UPS with your exact PSU. Of course, a pure sine wave UPS is the safest. My guess is that a stepped wave UPS would be fine for the vast majority of PSUs.
I am personally going with the Corsair 1500VA sine wave UPS at $199 vs it's non-pure sine wave UPS at $139 ($60 premium just to play it safe).
Good luck to you.