I assume you mean A firearm when you use the euphemism "protection". The current Department of the Interior policy is that you have the same rights for firearm possession within National Parks as you personally do within the state within which the park is located. BLM policy is the same.
Attacks by 2 footed predators is exceedingly rare in wilderness areas, as are ones by the 4 footed kind. A bear will be hardly dissuaded by anything less than a +P 45 FMJ. A 9mm shot, unless very well placed, will be ineffective against any large predator. A hollow point, designed for people, will not have the penetration needed for a bear. A shotgun with a rifled slug will work, but I doubt that would qualify for SUL status. I know of bears being brought down by a 22 mag. rifle. It is quite conceivable that the sound of a gunshot will scare a bear off, a pistol or revolver shot is very loud.
So the question is, are you a good enough marksman, who will stay highly focused in a moment of extreme crisis, to make a killing shot to a running animal before it closed from 25' in the 1.5 seconds you have? You would need to stand in the line of a charge, and calmly place multiply shots within the 9 ring of a rapidly approaching target that is bouncing as it runs at you to kill you. You will not get a side shot at it's temple or ear cavity or spine. It will need to be a shot into it's open mouth or base of skull or eye socket. If you hit a bear with a 9 mm between it's eyes, the shot will most likely bounce off. If you make a heart shot, you still need to wait a minute or so for the animal to drop. Can you even draw your weapon in that time? Do you practice draw, present and fire enough to be highly proficient and accurate? But on the other hand, a rabid raccoon was killed by rangers in the Grand Canyon at the Phantom Ranch campgrounds a few years ago. It took them far too many shots to kill the poor thing.
I have a carry permit, practice often and have carried on solo hikes. I have met rangers while I carried openly. Since it was in AZ., an open carry state, I was legal and they did not comment, which was proper, I was doing nothing wrong or illegal. I am not sure I could defend myself from a charging bear, I most likely would not hear a Cougher until it was in the air or on me, but I could defend you if you were attacked, but if I were solo, that point is mute. Could I draw once I was attacked, hard to tell, depends on my wounds. There are reports of people successfully fighting off predators. So perhaps I could, and a pistol is hard to borrow in an emergency. Perhaps the pain of several non-lethal shots would convince a predator to end an attack if it was meal based.
In my youth I carried condoms on solo hikes, just in case, but never got to use them. Another fantasy unfulfilled, so sad!
Edited for that speling thing.