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Yep, if you want to dink with it the scope would be the only thing I would consider upgrading.
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This. Ruger American should be a very accurate rifle and .270 Win is plenty of cartridge for whitetail. Bushnell Banner will be perfectly serviceable for hunting.
I used a cheap Simmons scope (a step down from a Bushnell Banner IMHO) for two years on a 12 gauge slug gun, and killed deer with it.
But, if you want to spend some money I'd upgrade the scope like dbd870 says. For less than $200 a Vortex Diamondback would be a noticeable step up in glass quality.
Make sure you go out and practice with the rifle. I'd buy a few other brands of .270 and try out a box through the rifle. Find out what your rifle likes.
You don't need the latest super technology, polymer tipped, ultra expansion bullet to kill a whitetail with a .270. Decent soft point bullets will do the trick.
I'd recommend trying some Prvi Partisan PPU in your rifle. I've seen their soft point ammo shoot very well, and the soft point bullet they load is of good quality. I used PPU 6.5x55 to kill three deer last year without issue.
Good luck!
ETA: Once you've gone hunting and if you decide it's your "thing" I'd still hold on the rifle. I started hunting later in life myself, found I loved it, and then (much like BRD) found I had a case of DRD (deer rifle disease).
If I'm travelling any distance of a trip to go hunting I always bring a backup rifle/bow/muzzleloader in the truck. Nothing worse than some problem with you weapon cutting your hunt short.
Right now you've got a solid entry rifle for hunting, and in the future it will make a great backup rifle.