With Bear, for defense, not hunting, penetration is the key. Penetration into a central nervous system branch-brain or spine. In theory a heavy 10mm slug can do this, it is a great penetrating round. A hard cast 200gr FP should do the job. Lots of folk are touting a max load of something bigger because it IS hotter, but maybe not so good on repeat shots. Bear have been STOPPED with the lowly 30-30, and even weaker rounds. Of course the 30-30 is better than any handgun round-if it has the right bullet.
One must remember that the Bear might be a 800 pound critter moving at 35 MPH, humping and swaying all over the place. How do I know? Been there, done that. Bear when attacking have a very annoying habit of throwing their head back as they charge, thus not giving one a clear shot at the brain pan. Yeah you can shoot them through the mouth into the brain, but it is much better to get a solid hit on the front top of the head.
It takes a cool head and a steady hand to stop a Bear, that is Mister Bear to all of us.
If a 10 mm was all I had,I would have it loaded with minimum 200 gr flat point FMJ or hard cast bullets, with 225 being better. However I would really want my 45/70 or .458, even a 12 gauge with hard slugs is better than any handgun.
I have just finished deconstructing a Browning BPS 10 gauge 3.5" magnum, 19" barrel, open choke, rifle sights, and 1.5 oz hard cast slugs, it might do the job, or it might launch me out of Bear range.