As this graph from The Economist shows, the correlation between gender inequality and democracy is quite strong, but it is not a perfect indicator. Singapore is a country with gender equality on par with France but is as democratic as Turkey. On the other end, India aligns with France on democracy but gender inequality sits as high as Saudi Arabia.
The slope of the line of best fit does show a rather strong correlation between democracy and inequality, but advocates should not jump too quickly. It also does not do anything to hint at causation. Could it be that greater democracy leads to lower gender inequality? That seems to have been the case in the United States, but I am sure that argument could be flipped on its head pretty easily. The same goes for all the other nations. There are strong Singapore says you don't really need democracy that badly and India says that gender inequality does not matter all that much.
Maybe it is time for Hans Rosling to bring this graph to life!