I understand the argument that firsts are special, but I think it's important to bear in mind several caveats here:
1) It's a group of 6-8 people, won't there be any other women? And if not, maybe that's a bigger problem.
2) This 'first' woman to 'go to the moon' isn't actually stepping on the moon, and when the first woman does finally step on the moon, that will surely overshadow this billionaire's date (consider, how many people can name any astronauts on Apollo 9/10 off the top of their head?).
3) Do you think anyone has the right to tell this guy how to spend his money? Without his money and his extravagant use of it, the trip wouldn't happen, so I think he reserves the right to take any guest he chooses (however he chooses). And if you do want to argue about how people spend their money, you should probably be arguing that they spend it on charitable donations or scientific progress (of course in one sense, paying for her spot is paying for scientific progress), not over what guests are taken on a tourist trip.
4) Do you think the 'first woman to go to the moon' should be a qualified astronaut who has been given a free spot on a tourist flight? I don't think that's very different from a highly qualified (female) biologist/zoologist, being given a spot on a tourist safari in lieu of a real research expedition. In my mind, that's far more insulting to her than having her wait a bit longer until a real trip that properly utilizes her skillset can be afforded.
TO CLOSE (this unnecessarily long comment), it's been 45 years since NASA has landed on the moon. Rather than getting upset by how very wealthy people spend their own money, maybe the point people (who are upset about this) should be making is that NASA should go back to the moon and take qualified women astronauts (not for the sake of taking women to the moon, but for the sake of conducting valuable experiments with state-of-the-art technology and taking women is only a natural part of fulfilling that objective).