My termination was based on false allegations. Do I have any legal recourse

I am an ultrasound tech with over 40 years experience. On the day I was fired I learned that 4 patients had accused me of touching them inappropriately during their ultrasound exams. I didn't. If anything "improper" occurred, it was accidental and certainly not intentional. In my opinion, when the company received these complaints, it could have conducted an investigation or it could have had another tech randomly sit in my exams to observe and identify any actions that could have been misinterpreted. I would have fully cooperated with any such courses of action. Instead, the company opted to fire me, which was probably the easiest thing it could have done. I need to look for another job now but I am worried about filling out applications because, in a face-to-face interview, I will have to explain why I am not eligible for rehire. I thought about saying something vague like, "I had a disagreement with my employer" but I'm sure I would be pressed for details. The allegations alone, even though they are false, are so horrendous that I worry any prospective employer will be scared off and show me the door. Do you have any comments or suggestions? Thank you.