You might have a point if it weren't for the fact that it's only due to religion that the US recognises the Vatican as a state in the first place. The UN doesn't. It's only a member on a special case basis, not as an actual state.
The UN doesn't because the UN's membership doesn't want the Vatican to be a state because they're not Christian? While true, it's also not true in the way you'd like it to be. They believe that by withholding recognition in the UN they are denying power to the Roman Catholic Church, not that the RCC does not merit it.
Nearly all of Europe recognizes the statehood of the Vatican, even the parts that are strongly Protestant. The Vatican has a proper ambassadorial system, the Papal Nuncios. They are treated in nearly every country as an ambassador would be, even China and back in the bad old days the USSR did so. Withholding recognition of statehood in the UN is a fig leaf to hide behind in this case. The Pope wields a public power in his sphere comparable to most heads of state. (Which is to say, not very much.)