This is a tough one. It is always “deeper” than what is revealed publicly; we’ll never truly know what goes on at the diplomatic level.
One angle I have is this: The White House wanted the Pope’s approval on this in a VERY major way. They have sufficient access to put their “best reason/case for war” on his table. I am sure they have tried their very best. JPII is not approving for who knows what, and for how many reasons? It is possible the Holy Father is wrong, but if they had presented some real major-compelling evidence, why would he say no?
Prayer (as always) is the best we can do.