And the relationships he shows really are impressive and somewhat surprising, but it seems to me that he screwed the pooch on this one. I say this, and am fully aware that I'm no statistician, but I'm seeing methodological flaws everywhere in this study that a first-year student should easily be able to pick up on. See if you can spot those problems:
Even if I didn't dispute the conclusions he draws from the data he uses, I would definitely question the validity and reliability of said data. Many people, for instance, and especially in developed nations, claim membership in particular religions only nominally. In such cases, religion almost certainly does not play any kind of causal role in their family planning decisions. But when you include these people with those for whom religion actually is an important motivating factor, you wash away the actual impact of the latter's religious beliefs and the causal influence on those choices. If there were some reliable way to distinguish religious believers from religious nominalists, I would not be surprised if the results of Rosling's studies changed quite dramatically.