Despite the general agreement that the old man is an angel, letters from Rome show disinterest, and even a certain amount of mockery. It seems that the Vatican is not convinced, and so they delay until they stop receiving letters, not wanting the expense of sending a high-ranking official out:
...the mail from Rome showed no sense of urgency. They spent their time finding out in the prisoner had a navel, if his dialect had any connection with Aramaic, how many times he could fit on the head of a pin, or whether he wasn't just a Norwegian with wings.
(Márquez, "A Very Old Man with Enormous Wings," salvoblue.homestead.com)
This seems to indicate that if the man is indeed an angel, his presence on Earth is not at all interesting to the Vatican, symbolizing a possible need on their part to be the sole arbiters of religion. If there were a truly supernatural being on Earth, the Vatican would lose quite a bit of its power, especially if the angel gave out information that contradicted the Vatican's stances. Therefore, since they don't want to be superceded, they wait until human nature loses interest in the angel, and it leaves on its own; the delay is a symbol for both indifference and dismissal.