Why doesn't this paradox disprove (some) multiverse quantum gravitational theories? As I understand it, one theory of the multiverse is that there are an infinite number of universes separated by small distances in other (than our observable 3/4) dimensions and that gravity is weak because it permeates these other dimensions.

But if there are an infinite number of these universes then surely the gravitational force we would feel would be infinite?
The answer to your question for brane cosmology is simply that gravity follows the inverse-square law. We don't feel the gravitational pull of, say, Mars because it's so far away. Any other universes are so unimaginably (all right, that's hyperbole) far away in comparison that any gravitational effects by the matter in them would be very, very, very, tiny.

A similar argument might be to say that an infinite series always goes to infinity because you keep adding on terms. But this is often not the case, as with the series
$$\sum_{n=0}^{\infty}2^{-n}=2$$
It is infinite, but it converges.