To make headway in a suit against the employer, Bob has to establish that the employer has a duty to not fire him. In the US, firing could be illegal as discriminatory, but being an alleged animal abuser is not a protected class. There are other forms of wrongful termination such as retaliation for a legal action by the employee (reporting a wrong-doing or filing a workman's comp claim). In California, Connecticut, South Carolina, and Louisiana, political expression is a protected activity but otherwise you can be fires for being red in a blue shop. Montana does not adhere to the at-will employment doctrine, so after the probationary period, you can only be fired for good cause which
means reasonable job-related grounds for dismissal based on a failure
to satisfactorily perform job duties, disruption of the employer's
operation, or other legitimate business reason. The legal use of a
lawful product by an individual off the employer's premises during
nonworking hours is not a legitimate business reason, unless the
employer acts within the provisions of 39-2-313(3) or (4).
Elsewhere it reduces to a contractual question. As a tenured professor in the state university system of Ohio, there are specific criteria for firing Bob, which do not include being an actual animal abuser. So Bob might prevail in a lawsuit for damages against the university. But as a programmer working for Microsoft, pursuant to para 2 of the employment contract, Bob can be fired at any time for any reason. As a star player for the Seattle Supersonics (cough), there is a specially crafted clause in the contract that only allows termination of the contract before the end of the 5 year term for "egregiously immoral behavior". In that case, Bob would likely prevail in the lawsuit since he did not engage in egregiously immoral behavior, by any reasonable understanding of the expression. Barring such a contract clause, Bob is out of luck in a lawsuit against the employer.