Our biggest bears are akin to the lighter (but still very dangerous) African game.
You'd use a .338 Win mag, .375 H&H or the like. Or a short .45/70 lever.
A 12ga. stuffed with slugs is also good bear repellent.
The replies above by Mark8 and Autonomous are spot on.
Going into big bear country without a proper long gun is like all the Hollywood movies where the good guys storm a warehouse and all are armed only with handguns.
If any of the bad guys have a rifle, smg, or shotgun, the good guys are going into the fight with a severe disadvantage.
Same as in the backcountry, carry a hand gun for a last resort, but don't stray far from the gun with the real power, the long gun.
Most of my time when I wasn't active duty, I lived in SW Montana.
I had/have an interest in all types of firearms, from BP to old west cartridge guns to modern stuff.
For about 10 years I owned a G20. I felt adequately armed down in the valleys where a black bear or Cougar might be lurking.
But when I'd go up into the mountains I'd carry the 629MG.
Show me a 10mm load, any load, that will penetrate as well as a hardcast 260-300 grain .44 Mag.
I did some rather unscientific penetration tests during the decade I owned the G20.
During a visit to the desert I saw an old Toyota sedan that had been abandoned and had been the subject of much target practice.
I had the G20 and M66 with me, and for both calibers had my hand loads with the Speer TMC Silhouette bullet.
With both the 10mm and .357, the bullet weight was 180 grains.
Velocity was 1260-1275 fps. for the 2 TMC loads.
I also had some 158 gr.Federal JSP in .357 and some 175 grain Silvertips for the 10mm.
I shot the engine block with all 4 loads, fully expecting the 10mm 180 grain TMC/FMJ to sail through the engine block and go deep into the hill behind.
No, just left a smear where the bullet hit. Both .357 loads cracked and entered the side of the engine block, but did not exit the one side.
I also shot into the door and bullets only made it through if they missed the window linkage bars.
Lets just say that I was not so impressed with the 10mm after my rather unscientific results, the two .357 loads did much better.
Nothing magic about the 10mm, it is a stretched out and slightly more powerful .40 S&W.
With any handgun for potential bear use, it's gonna be a brain shot or nothing. Forget pumping half a magazine from your 10mm into the bears chest while he has you, in the minute or two it takes him to die he will have done all he can to damage you.
Getting through the bear's thick skull is the goal, and from what I've seen, there are better penetrating handgun calibers than 10mm.