Assuming you were an employee at will, it's not illegal to terminate you because they believed, even wrongly, that you stole money. They can terminate you for any reason or no reason, unlawful termination excepted.
What you describe is actually a very common situation. Employer wrongly believes employee committed theft. Employer doesn't have to be right to lawfully terminate employee. Even if you had a contract that you would hold your job unless there was good cause to terminate you, their belief that you committed theft would be enough. Otherwise, employers would have to continue to employ people whom they believed were stealing. And that makes no sense.
The issue of what they tell other employers is more problematic. In some places there is a statutory privilege which allows people to make communications, one employer to another, like the one you describe. On the other hand, if you didn't commit the theft, it sounds defamatory. Except you can't prove you didn't commit theft, and there is circumstantial evidence that you committed it. So, in my judgment, if you bring a defamation claim, you lose.
I'm not licensed in DC so don't take what I say here as legal advice. It's simply my analysis in light of the general principle. If you need legal advice, please consult a lawyer who holds DC licensure.