Women have few rights in poor economies. They enjoy much more rights in rich economies. There is no doubt the correlation is positive, but what is the causation? Asking this question puts you on a slippery slope, as advocates of women's emancipation are quick to dismiss any hint that underlying economic forces could be the origin of new rights for women, rather than an exogenous political push. What does the literature have to say about this? Luckily, two surveys came out within a week of each other, with contrasting views.
Matthias Doepke, Michèle Tertilt and Alessandra Voena start from two observations consistently documented in the literature. 1) Giving women more rights refocuses private and public spending towards children, in particular health and education. Both lead to more growth. 2) Technological progress increased the costs of a male-dominated society and encourage females to join the labor force. Both lead to women's emancipation. And not put 1) and 2) together, and you have a theory of growth with female empowerment, where males are willing to step back for their own good. If technological progress initiates this virtuous cycle, it is actually not necessary once the cycle started spinning, and we have endogenous growth.
The other survey is due to Esther Duflo, who does not see anything self-sustaining in this cycle. In particular, one needs a constant exogenous push for women's equal rights to keep it going. While the paper above looks at long trends, the second focuses on natural and artificial experiments where the analysis is rather short-term. It is thus not clear to me whether it is really about growth or development. I am thus more convinced by the first paper and hopeful that the virtuous cycle will continue feeding itself.