Well, no. Women's liberation has an effect on birth rates. But it is far more likely that it's more about atheism and secularism than women's liberation.
Let's look at your table.
Lets pick the country that has the lowest gender inequality score, Slovenia, the country with least births, Singapore.
2.2% Eastern Orthodox
Let's look at the highest gender inequality score and highest birthrates, Niger.
Let's look at chad for good measure.
I am noticing a pattern. The highest birthrate and highest gender inequality countries had little to no atheists and were strongly Muslim. The lowest birthrate and lowest gender inequality countries were mostly Christian, Buddhist and Atheistic.
Considering how Islam and most notably it's post-modern interpretations have a focus on spreading themselves, and the fact that these countries both are in the top 3 worst HDI scores in the world (Niger at #1, Chad at #3), abortions, birth-control and even condoms are probably off the table.
> But wait! What about Israel?
Well let's look at Israel.
But this doesn't explain why Israel has a low gender inequality score.
No, it's not because the Jews are secretly manipulating the numbers, or at least, it doesn't seem like it.
No, the reason is that simply saying "74.2% Judaism" doesn't tell the whole story. Let's take a closer look at Judaism.
49% Hiloni (secular)
29% Masorti (traditional)
13% Dati (religious)
9% Haredi (ultra-Orthodox)
Especially considering how Haredi families have 6.7 children on average, is it really surprising that Israel is at 3.07 Births per woman according to your research?
tl;dr: The religion of a country seems to be the cause of both it's gender inequality and birthrates. Therefore, they are related but not do not cause each other.