Many 32-cal revolvers will fire 32 Automatic ammunition. I know mostly of 32 H&R Magnum revolvers, but some 327 Federal and 32 S&W Long revolvers will also fire 32 Auto. The 32 Auto is a semi-rimmed case, so depending upon the exact dimensions of your cylinder, they may hang on and fire just fine. My own 32-cal revolver (a 327 Federal) will only fire 32 Auto cartridges a portion of the time, and some of those are light-strike, squib-like firings, so I don't try that any more.
As to pressure, SAAMI MAP for the 32 Automatic is 20,500 psi via piezo transducer (15,000 CUP via the old copper crushers). The SAAMI MAP for the 32 S&W is 17,000 psi, and the 32 S&W Long is 15,000 psi via transducer (both are 12,000 CUP via the old crusher method). 32 Auto cartridges are therefore not quite as high pressure as 32 S&W proof loads, but they're much of the way there. So, personally I would NOT make it a habit to fire 32 Automatic cartridges in a revolver labeled only for one of the 32 S&W cartridges. Between the higher pressure and the possibility that the S&W-labeled gun is older or otherwise metallurgically uncertain, I wouldn't take the chance and I'd recommend others don't, either, for their own safety, the safety of those around them, and the safety of their gun.