The nerve! Maybe you should join United's Red Carpet Clubs since Delta is...
the only VIP club in the United States which does not welcome its members when flying another carrier.
Oh, wait a minute. Never mind. From the Red Carpet Club membership terms and conditions:
Red Carpet Club reserves the right to limit Clubroom access to members traveling on same-day United Airlines tickets.
Granted they rarely enforce this, but that's mainly because the RCC's don't have free booze, IMHO. Regardless, that's United's policy too, so Delta is not the only one with this restriction and I wouldn't be surprised if it weren't actually the norm.
RCC's you usually just flash your card as you pass by the receptionist. I've found at CRCs they usually want to look at your card and ticket before letting you in.