1) What was the compliment? Some should be harmless, while others cross a line. If it was along the lines of "nice profile" or something similar, no big deal. If you complimented her figure, maybe your girlfriend is on to something.
2) Was there any further contact between you and the other girl, or was it a one time thing? How long ago was the conversation?
3) If your girlfriend is that insecure, you need to take some time and ask yourself if she's the right girl for you. I know that's not what you want to hear, but she obviously is down on herself, and doesn't trust you. I think it's a huge, huge breach of trust to go through your inbox.
Out of the two of you, I put her more at fault. From personal experience, people with those type of trust / security issues don't magically get better some day.
I mean, if you two really love each other, a harmless compliment to another woman shouldn't be enough to threaten an engagement. If that's happening, then there are obviously deep-seeded issues; either there's a reason she doesn't trust you, or she's got some issues of her own to work through.