The wage gap isn't evidence of anything other than men statistically tend to earn more than women. It's more an earnings gap than a wage gap. It is not in any way suggestive of inequality in opportunities. It's certainly something to be studied, and it has been studied. Thus far, comes down to three things:
1) Certain types of jobs pay more than others. As a society, we may want to question why an investment banker is worth more to us than a nurse?
2) Individuals make choices that influence their earnings. Women as a group tend to make choices that lessen their earnings.
3) When adjusting for comparable roles, the differences decline dramatically to low single-digit gaps.
There is no systemic segregation here. It seems more a case that men and women generally have differing drives. This kind of makes sense when you consider that evolution has been shaping us for millions of years to be successful reproducers. Strategies successful for male populations are not necessarily successful for female populations, and these drives go far beyond the simple act of reproduction.
Actual evidenced discrimination should be dealt with. Discrepancies between demographics should be studied, but a faulty premise is not a good foundation on which such studies would rest.