I think they were both pretty tacky, myself. The LW didn't actually refer her friend to the gym and was essentially asking her buddy to help her scam the gym. The point of referral rewards is to bring in new customers who wouldn't otherwise be going to the venue. If a potential customer has arrived at that decision on his or her own, then the friend has not provided a referral.
This is somewhat analogous to lying to the waitress about it being someone's birthday so he gets a free sundae. While it's not as sinister as taking advantage of a person, you are using deception to get a benefit from an entity that would not be giving it to you without that deception. While I do believe that people should give gifts with no strings attached, I also believe it's wrong to solicit them. I don't really see this as retaliatory rudeness, but I do think the LW is kind of calling the kettle black. Furthermore, LW's friend told her up-front that he wanted to share the benefits. She should have redacted her request at that point if she didn't like it.
Now, if the LW had actually suggested that her friend join the gym (arranged for a trial membership, brought her in as a guest, etc), I would believe that she was entitled to the benefit. However, she technically didn't do anything to deserve/earn a free month.