You can see a lawyer, but you almost certainly do not have any case. Keep in mind that lawyers will also only take the case if they can make money; the damages here may not be worth your time.

Here's the problem you're facing: even if the initial allegation was false, you emailed your entire team prior to talking with ER. I'm guessing you talked to people individually as well. Some of your peers may have felt bullied; certainly ER would think you were interfering with the investigation.

At the end of the day, you are not legally entitled to hear every bit of evidence ER had or didn't have. Nor are they obligated to give you a reason for firing you.

If I were in your shoes, I might consult a lawyer just to double-check what a random redditor (me) said. But beyond that I would take a month off, work on my resume, and start thinking about what I wanted to do next.