Hello Restaurantlady and welcome to JustAnswer.
In most states, employment is referred to as "at will", which means that with only limited exceptions, you can hire and fire anyone you want for any reason you want.
The exceptions would include termination based on one of the so-called "suspect classifications" (gender, race, religion, etc.), or where there is a labor agreement or personal contract
the terms of which define the circumstances under which employees
can be terminated.
Therefore, failure to reveal registered sex offender status, particularly if asked during the hiring process, would be appropriate grounds for termination.
If you let me know what state you are in, I can tell you whether or not it is an "at will" employment state.
Thanks for asking your question here on JustAnswer. If you have any other questions, please let me know.
Edit: Your optional and already tried information showed up after I posted the foregoing, so I now see that you are in Ohio, which is an "at will" employment state. Therefore, what I said above applies to your situtation.
Therefore, if you terminate this person now, he would not be able to successfully sue you for wrongful termination.
On the other hand, unless you asked him during the hiring process if he was a registered sex offender and he failed to answer the question truthfully, he would be eligible for unemployment insurance
benefits if you terminate him now.