DIANA
asks
GAGE
why he needs to pay for sex. He says that he doesn't and that he saw her as a challenge.

Besides the good points others have made about Gage having money yet is so lonely, and his attraction to Diana he also did it to see 'if he could'.
Diana had said that no one, including herself, could be bought. Gage decided to challenge that, and it worked.
I'd say this cloud is Cumulo Nimbus.
Didn't he discover America?
Penfold, shush.

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Doggy dolphins?
Erection?

he probably paid for some freaky stuff for a million dollars

Not everyone thinks Redford is the end of the earth, or even that handsome. And if Richard Gere had to pay for sex in Pretty Woman, Robert Redford definitely had to pay for sex in this movie.
Jack's not dead! Jack would never die without telling me, first!

He has money and all that it can buy, but he doesn't have the one thing that most people take for granted: a partner who loves them.
You're right. He says the very thing to David through the movie. Diana also asks the same inferred question as the OP, but Gage refutes the claim and says he's more interested in the psychological responses to his proposal.

I think it illustrates the character's unfulfilled life. He has money and all that it can buy, but he doesn't have the one thing that most people take for granted: a partner who loves them. He even says it to Woody Harrelson while they're playing pool before he asks for one night with Diana. Despite all he has he's lonely and unsatisfied. Sure, he probably has women lining up to date him and sleep with him because of his money, but it's likely the women he really admires and wants to be with are the very women he never ends up being able to have.