Can I start a wrongful work term case after being termed for something that was not my faullt

I worked at an assisted living place as a concierage front desk person and was termed after a resident came out of the Egis or memory care section by himself and passed the nurse who was supposed to be watching her patients. He was lead to my desk by a dining room person and then I waited for the nurse to come an get him - they said I should have left my position to bring him back. I have worked there 10 months, never one problem and have worked through covid and working double shifts and covering all that have taken off and never missing a day of work. There was no verbal or writtne warnings at all. I am looking for financial and not looking to go back to work there. I already have a letter of referral.