In such cases, evidence plays an important role. As you mentioned, he is only suspicious, and there is no evidence against him. Therefore, taking any action against him would be unlawful and unethical, setting a wrong example for existing employees.
Since you are involved in contract staffing, I understand the client's expectations and pressure from them. I would advise you that if the concerned staff is truly not guilty in this case, you may consider transferring him from one client's location to another. This way, your client will be satisfied that the suspected person is no longer working on their premises, and you have taken action on their request. Additionally, the concerned employee will not feel unjustly treated.
However, if you still wish to terminate him, and you do not have any evidence of the incident, you may request a resignation letter from him. Treat the case as a self-resignation and close it with an immediate full and final settlement of his dues.
I still believe that following this process may be seen as unjust to him. If you aim to identify the actual culprit, consider filing a police complaint as they will be able to uncover the truth.
Regards,
Tushar Swar