Assuming non-theism, a global nuclear war in the 20th century would have been a reasonable prediction to make (and indeed some made it)—but a prediction that was falsified. On the other hand, given theism, such a war would be unlikely. I think there is an argument for theism here. Let me say a little more.
Given naturalistic assumptions about human behavior, there were several approximately independent occasions (whether a short crisis or a longer stretch) during the Cold War at which it was moderately likely—say, probability 1/2—that a global nuclear war would result. Let's say there were four such occasions. Then, given naturalistic assumptions about human behavior, the probability that global nuclear war would occur would be about 15/16. On the other hand, given theism (and especially given Christianity or Judaism—specifically, the promise of the non-repetition of the vast destruction of human life in the Flood), the probability that global nuclear war would occur is much smaller, let's say 1/8. (It would be an order of magnitude harder to come up with a theodicy for a global nuclear war than for such horrors as are already hard to come to come up with a theodicy for.) Well, these two conditional probabilities show that non-occurrence of a global nuclear war incrementally and significantly confirms theism over naturalism.
If initially theism and naturalism are take to have equal probability 1/2, and we plug the above numbers into Bayes' theorem, we get a consequent probability of 0.93 for theism given the no-global-nuclear-war evidence. If initially theism is taken to have probability 0.1 and naturalism 0.9, we get a consequent probability of 0.61 for theism.
Of course, I've made up the numbers about the probabilities of global nuclear war, though they don't seem crazy to me. A historian would be needed to give us better numbers.