I recently resigned from my company. I gave three weeks notice and offered to work one day a week after that until they were able to hire a replacement. I recently discovered that the executive director has been telling people (including members of the board of directors) that I did not give any notice and that I broke an agreement we had that I would stay for one year after accepting a promotion (we had no such agreement and when I accepted the promotion, I told them point blank that I was unwilling to make any commitment as to how long I would stay with the organization).
What are my legal rights in this case and do I have any recourse to stop him from making these unfounded accusations?

What damages have you suffered because of these statements? What jobs have you been turned down for? What money have you lost?

None yet. But I work in the nonprofit sector, which is an incredibly small community and I'm concerned that this will impact me negatively in the future. I don't want any money or damages, I just want the director to stop giving out false information about me.

If you have not suffered any damages yet, you'd be wasting your time filing any suit now. When and if you suffer any monetary damages that you can directly tie to these allegedly false statements, THEN you can take legal action. Until then, you've got nothing to go on. You can't sue for what might happen.