Is this woman your coworker or your supervisor? If she's just a coworker you can do things to avoid her. If she is your supervisor, you can't really avoid her.

I've read some studies that recommend not personally disclosing information to your coworkers for the exact reasons you've written about: your new hire. Now, your coworker, who didn't really need to know about this new hire, is being her nosy, intrusive self with you, and undermining your work flow with her shenanigans, and basically creating havoc for you because she enjoys doing that.

If you don't have to work with her, then don't. Are you required to work with her? Can you have her transferred or can you transfer so that you won't have to work with her? What does your mutual boss' role here?

I would just never open up to her again about anything and just act all business-like on the surface and eventually she'll pick up the cues from you that you are distancing yourself from her. If you confront her, only bad can come from it. She would take the confrontation to your mutual boss and get you fired for it. So, don't do that.