From experience I'd say the airbag ECU has detected an open circuit, i.e. no connection or really high resistance, thrown an error and given you a light to look at.
It's highly, highly unlikely that the airbag has decided to die all on its own for no reason, it just doesn't happen. They're stupidly simplistic bits of kit. Essentially a little ignitor coupled to a small explosive that when set off fills a bag with gas. All an explosion is by definition is a rapid expansion of gases.
If you're really unlucky it might be the airbag ECU has gotten damaged, either heat or moisture, a dry join, failed POT or something and gone bad. I've known that happen.
More likely is a poor connection. Given how easy they are to disassemble, pop the bag off and check the connections on the clock spring and the bag itself. Give them a spray with some contact cleaner and reassemble.
I'm not sure off top of my head what system Fiat uses in the Stilo, it'll likely be either the MY97 or MY99. Probably the later 99 by this point actually, 97 is like pre-2000 I think.
Its been a while so I can't remember off the top of my head. One of the systems will 'clear' itself, the other won't. So don't be immediately surprised if the light doesn't go out my itself. I think it should but it's been a while since I'd had to deal with one. If it won't clear itself it'll require reset by diag software.
MultiECUScan will do this but you need to connect to the correct pin on the diag OBD port. I assume you've got this covered as you're aware of your fault codes.
They usually scan themselves on boot, i.e. ignition on, so if it's going to go off it'll probably happen a few seconds after you turn it on and start the engine. If it doesn't go out after 10 seconds, it won't go out.
Disclaimer time - work with airbags is playing with explosives and care is required. Frankly, I've work on airbags with batteries connected, ignitions on and even engines running before now; you'll struggle like hell to set one off but if you do you'll regret it.
Play it safe and have the battery off whilst you tinker.