I had left a private chat open at my computer complaining about a coworker of mine and she read it. I had complained that she told another department some incorrect information. She confronted me and told me that I should talk to my manager if I thought someone was giving out incorrect information. What should I do? Should I talk with my manager about this and if so what should I say?
I wouldn't act on this. There's no point in bringing this up to your manager, unless your coworker decided to talk to them first. This is fairly minor and not something worth bothering management with.
Next time, don't leave your chat open when you've been complaining about someone. Or better yet, don't complain about a coworker somewhere that leaves a paper trail.
Your colleague was correct in asking you to take it to her manager, although it would've been better to pretend she never saw it. But that's done now.
It's generally a good idea to never write something that you wouldn't want the other person to read. Not only does it save you from situations like this, which are vanishingly rare, but it forces you to think the situation through and come up with a better course of action.
You might invite her to a lunch/coffee and apologize to her for complaining about her in the chat and try to be friend with her. Lots of friendships start at such point.
Next time, when you had such problems, you may try to speak to the person, rather than to take it to any third person, including managers.
While it is stupid to leave your works computer unlocked with your private chat on display for everyone to read, that doesn't give anyone the right to actually read it. Nobody should ever read what's on your screen, because that could be highly confidential information - for example highly confidential information about a customer that nobody other than you should read. So your colleague is very much in the wrong here.
You can tell her that. Whatever complaint she has, her reading your private chat (during her own worktime, when she should be doing her work anyway) is the worse thing, so you can tell her that straight away and don't accept her complaint at all.
If she wants to go further with her complaint, you deny everything and ask her for evidence. Since you are not going to show your private chats to anyone, she has no evidence.
Note that I'm talking about private chats. Anything that she can read from her facebook account because you posted on facebook is not private. In the UK, people have been fired for posting on facebook where anyone could read it.