Perhaps. If one gets lucky. But larger and heavier calibers are recommended. Try .338mag for starters. Or .375 H & H.
'06 should be enough for most black bears, though. Brown bears can get about 3x bigger.
Back before the animals all got body armor, the old '06 was plenty big medicine. My brother killed everything in Alaska with one, my Dad used the .308.
I got caught up in the magnum craze. 308 Norma Mag, 350 Rem mag, even a 458Win. Even had a 338 mag. Hated it! Sold it back to the feller I bought it from.
Then, I bought a .280 Rem. For the first time in years, I really enjoyed shooting, I could even see the muzzle blast through the scope. I didn't need a spotter to tell me where I hit, I could see the bullet make a hole in the target! The "06 is enough gun for ANYTHING on the North American continent. Shot placement is everything.