I think the problem is the girl not you. If you asked her politely, she said no, and then you backed down, there is no reason for her to bad mouth you to friends. I could understand maybe asking one trusted person for advice but spreading rumors through a friend group is a serious issue. Not just the girl but also the friends spreading the rumor. I think a lot of people are really bad at communicating and it leads to more issues. If she had simply talked to you about how she felt, I’m sure you both could have worked through it.
The retirement community my elderly uncle lived in worked with a local hospital that would schedule all the appointments for the same day, around the same time. They had almost every specialty in the same building so the residents could see the various doctors they needed. But if you had your own doctor at a separate place, you could schedule it ahead of time and they would drive you there. However this was a high-end community that cost a lot, so I’m sure if cheaper places would offer the same options.