I think that's exactly backwards. Afghanistan has the highest birthrate. The less gender equality, the higher the birthrate. You get astronomical birthrates in the Hasidim and Islamic parts of Africa where women hold a subservient place to their husbands and have a constrained gender role. The Nazis had some success raising the birthrate by promoting and elevating stay-at-home motherhood, not by promoting equality.
I think there's a strong case to be made that banning women from colleges and professions would dramatically raise the birthrate. I'm not advocating this, I'm just saying it's what the data suggests. The more professional status is available to women the less they reproduce. This is very clear all over the world across cultures.