There are laws on what you can say. Its not allowed to insult someone. And its not allowed to insult ethnicities. Basically the same like every other freedom right: it end where the freedom of another person starts. So no, it wouldn't be possible to ban any mention of gayness.
On the other hand, laws don't fully protect you from this. The US has a very wide understanding of freedom of speech and at the same time has a 'don't say gay' bill.
Thats not the reason, but economics is. For example Korea and Japan – where the gender norms are pretty conservative – have a lot lower birthrate than Sweden – one of the most equal countries. It more depends on wealth being the most important factor, social security systems and workforce participation of women.
So either avoid getting wealthy, get a lot of immigration or get the social security systems in place to make it easy getting kids.