Now we're talking seriously dangerous game. Shot placement is always the single most important criteria, so while I consider a .30 caliber the absolute minimum, you're better off with a .30 caliber
you shoot well than a .416 you're afraid of. Somewhere in the middle is probably the best choice.
Hunting the grizzly bear and hunting the Alaskan brown bear are not the same. Grizzlies are somewhat smaller and are often hunted in high country where shots out to 250 yards might be called for. The larger Alaskan brown bear is generally hunted in 'tighter ' country. None of us has taken many of these bears, so I freely admit my choices are based on conventional wisdom as well as personal experience. For grizzly I think there's nothing better than a fast .33 — the .338 Remington Ultra Mag. or .340 Weatherby Magnum. But if the recoil is too much — and it might be — tone it down to the .338 Winchester Magnum. Because range can be involved, a 3-9X scope is a good choice. For bullets, stick with 250-grain slugs with known penetrating capabilities. I used a 250-grain Nosler Partition on a couple of grizzlies. That's still a benchmark bullet, but all heavy-for-caliber 250-grain slugs will do the job.
You could use the same setup for brown bear and you'd be just fine, but I think this is one of the only situations in North America where the .375 H&H is still king. But if you prefer a .375 Ruger, that's fine with me. And if you can handle the added recoil of a .375 RUM or .375 Weatherby Magnum, that's maybe even better. Skip the 270-grain bullets and use a 300-grain slug, then hope you run into a monster that's twice the size of any grizzly. Choose serious bullets: Swift A-Frame, Barnes TSX, Hornady DGX, Nosler Partition.
The challenge with the .375 is that all too often we saddle them with a low-powered 'dangerous game ' scope with maximum power of 4X or 5X. There is much potential for a 200-yard shot, which any .375 can handle. But don't make things more difficult by going 'minimal ' on magnification. Consider a scope with an upper level of 6X, 7X or 8X. Keep it turned down unless you need the magnification and you should be ready for any shot that comes along. By the way, brown bear country is pretty wet, so this is the place for synthetic and/or laminate stocks.
As you've undoubtedly gathered, I'm mostly a bolt-action guy. Call it a generational thing. Realistically, my big bears are probably all behind me (unless I hit the lottery and go for a polar bear). All have been taken with bolt actions, but in my experience all bears are tough, and the bigger the tougher. Clean, one-shot kills aren't the norm. So if I were to start over, I would definitely use my Blaser R8 with an appropriate barrel. The straight-pull action is significantly faster and requires less movement, and on big bears this might be critical.