I carry a .38 snubby in the woods. If a bear comes after me, I am screwed no matter what I have. Bear are not just meat eaters, this is what makes them so dangerous. A mountain lion cannot afford even the slightest injury since he cannot hunt with a bullet wound in his abdomen. A bear on the other hand, can take the wound, still track you down, kill you and eat berries and insects for the next month or so while he heals. If you see a bear in the woods, don't try to shoot it, try to move slowly and quietly away from it. If it spots you, just stand there like an idiot because running will spark the instinct to chase. If it does charge, I have no good advice, maybe laying in the fetal position crying for your mom will make him sympathize with your defenseless situation
Full blown meat eating predators will not risk great injury to eat. They would rather give up and try again another day. Usually, just the sound of a gunshot will send most critters hightailing it into the woods.
Of course, this is just my opinion and if you try any of this and it doesnt work, tell your surviving relatives not to sue me.