Is there any legal recourse (for the fired employee) to compel the employer to support his claim that a theft occurred?


No.

The employer is free to fire the employee if he believes that the employee was involved with the money going missing. Even if the employer is wrong. The employer does not have to prove that there was money missing, either. You really have no valid reason, based on your posting to believe that there was not.

As to what the employer can tell others, that's tricky. If the employer says, Mr. Employee was fired for stealing, and doesn't have proof, then the employee should talk to a lawyer. But if the employer says, Mr. Employee was fired because of a suspicion of theft, well, he WAS let go because of a suspicion of theft.

And quite frankly, unless you are either the employee or the employer in question, this is none of your business. I take it you know about this from the employee's point of view? The employer might have a different tale to tell.