I'm going to respectfully disagree that this story was sexist. While it's true that the main character is sexist, that's deliberate, and a part of the overarching themes of this brilliant and subtle story.
This is the tale of the Great White Male Savior (a la Avatar or Dances with Wolves) completely subverted and stood on its head. From the get-go, it's clear to me that Gallinger is not a nice guy, and that we don't have to like him. He's so brash and arrogant that even all the other humans can't stand him. He (and the other humans) initially approach Martian culture and society with an attitude of cultural imperialism and condescension, treating them as if they are primitive and inferior to Earth culture. And yes, Gallinger's attitude toward the matriarchal Martian women is sexist and patronizing, especially initially. But the key to this story is that everything Gallinger believes and assumes is wrong, wrong, wrong.
He believes that Earth culture is superior to primitive Martian culture, only to discover that Martian society is astonishingly ancient and rich, and that Martians themselves are biologically superior in that they have incredible life spans and are therefore able to accomplish far more artistically than Earth people are.
He thinks he's a genius for figuring out the sterility problem, and pontificates on how he's going to expose this truth for all the researchers of Earth who failed to put it together, but never considers that only the Martian men might be sterile.
He rushes out into the desert to save Braxa in a stereotypical gesture of gendered romantic gallantry, only to find that she doesn't love him and never did, and only slept with him on orders. To add insult to injury, Braxa isn't even pleased about the fulfillment of the prophecy and that her child is saved from death, as she never believed the prophecy herself.
He mocks their religion, only to find out that in doing so, he fulfilled a religious prophecy and only strengthened the religion in the end.
He criticizes the Martians for their fatalism and nihilistic attitudes, and then attempts suicide himself.
Gallinger thinks he's a great poet and hero who is acting upon this story, but in the end he's just a pawn of fate and all the people around him. Nothing he loves or believes in pans out. The Martians get a happy resolution to their story, but Gallinger can't even die when he wants to. Gallinger is exactly like the force-grown rose: it looks beautiful and full of significance at a glance, but everything about it was cultivated or engineered artificially, and is less remarkable than it seems.