Depends on who she told. Yes she would have a responsibility to tell your manager (or anyone in his or her chain of command) if you said you would be resigning. No, she should not tell your co-worker, Steve, who has no supervisory reponsiblity over you or Mary in Sales who would have no professional reason to know.
If you feel she miscommunicated to someone in your chain was what said, then I would go talk to her again and explain what you heard she said and why that was not the impression you intended to create. It may or may not be too late for this, but I would at least try.
Remember in the future that HR is not your friend. They have a duty to protect the company not you. There are times when helping you is in the best interests of the company and you can generally trust them in terms of things like benefits questions and fixing the problem with your pay (unless you work for a highly unethical company.)
But in general, you have to consider that nothing you say to HR is privledged and it will be discussed with others in your immediate chain of command if the HR person feels it is needed. They are, however, supposed to keep from spreading casual gossip to those who are not directly involved in an issue.