No. Rationality has very little to do with it. How rational the participants are doesn't itself change the fact that there is a competitive labor market and that salary offers thus must be competitive.

If women choose to forgo higher salaries in favor of something else, that's a different issue and it doesn't immediately mean that they are being "irrational", it just means they favor different things than men do.

The rationality of the players is central to any argument you make in the line of "there is a competitive labor market and that salary offers thus must be competitive".

If the hiring managers (of any gender) have an irrational bias towards thinking that women are not as effective as their male counterparts, this will affect the respective pay scales even within a competitive market.

Note that I'm not providing evidence or making a claim about the gender pay gap. I'm just saying that the claims you've made have many possible consistent explanations, including irrational bias against women.