Assuming there is a god (for the sake of the argument. I'm not taking any stand in regard to that question. Yet), and assuming he's timeless, then the only logical conclusion can be that his existence is in no way related the existence of some earthly religious establishment and its political manifestation.
The catholic church existed prior to the establishment of the Vatican, it existed when the popes exiled from the Vatican to Avignon, so I don't think that the existence of the Vatican is really eternal.
Also, as far as I can recall, other manifestations of God's rule over mankind, which were considered holy, disappeared over time. The Holy Roman Empire ceased to exist, but no harm came to catholicism. Monarchs, ruling as God's representatives, were overthrown. The papacy itself was, for centuries, in no way a religious authority, but a mere position of power and influence in European politics.
To conclude: I think that the papacy and the Vatican state are well established, and I don't see when or why they will cease to exist. However, assuming they will someday cease to exist (as all man-made things are meant to), I believe that the belief in God among humanity will not fade, but perhaps alter in some way.