Phil Shoemaker killed a large male brown bear last August with a 9mm loaded with 147 gr Buffalo Bore hard cast bullets after it attacked a fisherman he was guiding.
Shoemaker is a regular poster on other forums and had done a lot of testing prior to this incident, he felt confident enough to leave his 44 mag at home. He has been guiding bear hunters and fishermen for about 40 years. He's killed, and seen a lot of big bear killed so he knows his stuff. I'd imagine the same load from a 357 Sig wouldn't do any worse.
http://www.wideopenspaces.com/alaska...-a-9mm-pistol/
On the other hand I personally have more confidence in a hot loaded 10mm. I carry DoubleTap 200 gr hardcast at about 1250-1300 fps from a G29 or G20. That load has proven that it will also stop big bear and I've seen video of a hunter taking a Cape Buffalo with that load.
Either way, as an outdoor load for larger animal protection you want heavy for caliber. I keep the 200 gr loads in my G29 when in bear country, but change to a magazine loaded with 155 gr ammo at around 1400 fps for 2 legged threats.