The question in my opinion is two-fold:
1) How over gunned do you want to be for whitetails?
2) How under gunned do you want to be fore elk?
Within these questions are a lot of other questions, like what will your typical range be for whitetails? How much recoil can you tolerate? What's your physical condition like?
Frankly, we all know we don't need anything more than a .243 Win or .30-30 to hunt deer to 250yrds, which is LONG for whitetail shots, so essentially, everything bigger than those are "overgunned".
However, a .243 Win or .30-30 aren't going to be first choices for elk (not even legal in some states for elk).
Personally, I've used 7mm RUM, .300 WSM, .300 WM, .338WM, .375 H&H, and .45-70 for elk. My current elk rig is a .300 WSM, my wife's elk rig is a 7mm WSM.
As others have mentioned, having a backup rifle is a good idea. I take a .30-06 sporter along on every big game trip I make, from deer to elk to blackbear to moose.
So, a 7mm RUM or WSM or .300 WM/WSM/RUM are pretty overkill for deer at 100yrds or less, but they work for deer hunting, and they're GREAT for elk out to as far as you can shoot. A .270win is great for whitey's to 250-400yrds, but might be a little light for a BIG elk at 650yrds... But it'd work for elk hunting...
Since this will primarily be a deer rifle, but will still see regular duty as an elk rifle, I'd recommend a .300WM or WSM.
Another really good option, which is cheaper than buying 2 full rigs, is to buy a switch barrel rig like a Savage bolt action rifle. The rifle itself will cost you $800, then a spare barrel, bolt head, and barrel nut wrench will run you about $450-600, and you won't need to buy 2 scopes. It takes about an hour to change barrels on a Savage.
My wife uses the same Savage rifle for prairie dogs, deer, and elk. She has a Savage 12 short action, shoots a heavy barrel .223rem for small game, a midweight contour .243 WSSM for deer, and a sporter contour 7mm WSM for elk, all using the same action, trigger, and scope (2 different glass bedded stocks, one sporterweight and one benchrest style).