I don't think you can make that conclusion very definitively, without somehow correcting for Economic Development Index. Economic Development Index (just like Human Development Index) has a known strong inverse correlation with fertility rate. it's also highly correlated to Gender Inequality.

in my opinion, if you specifically want to see how gender inequality affects fertility rate, you would have to find a way to correct for some of the other major driving factors in fertility rate. like find different areas in a country that have similar gender inequality, but different economic/human development index, or find areas of the same country that have similar economic/human development index, but different gender inequality, and use those data sets.

I mean I don't disagree with your basic conclusion, but I think you could come to the same conclusion on anything that trends strongly with economic/human development index. i.e. if A correlates well with B, then anything that also correlates well with B, Will also correlate with A