I think we have to pick apart the distinction between something being owed versus it being expected. Whilst men don't owe women anything, there are various expectations that exist that a man ignores at his (reproductive) peril.
The same can be said of women, of course, but it's a different set. These unspoken social obligations are more behavioural for men in that they're judged more for their character and capabilities, whereas women tend to be judged more on their appearance and femininity.
Of course, people can be who they want, so in that sense they don't owe anything to anyone, but whether they conform to these expectations or not will have an impact on their perceived worth as a person and as a potential partner.