If you haven't paid this bogus fine, do nothing.

They've accused you of something you didn't do, and you've refuted that claim in writing. That's all you need to do for now. Since it's via email, you've got copies. You can print them if you like, so that you've got a hard copy.

If they do later respond, refuse any communication via phone, even if they call you directly - just say that you'd rather communicate via email so that there's a record of everything said. Keeping written records is really important. It may also be a good idea to check review websites to see if anyone else has had similar complaints.

Chances are they won't bother taking any kind of legal action to recover the 100€ fine because, even if it were legitimate, it would cost them more in time and money to recover it. Even less likely if, like you say, it's a false claim.