Not a nice way of staying a night in Italy....

However, it will not be easy for you. I assume the case will be under under Italian law. I am not qualified in Italy but the systems in Europe are not that different but certainly very different from the US. For a beginning: There certainly is no general liabiltiy for accidents with equipment in a fitness room. If a piece of equipment brake and the business can make evidence that it was (a) standard equipment (b) properly checked and maintenance was performed, you will might have a problem. Certain accidents are considered to be bad luck (outside the US).

What I would do in your shoes: I would further try to escalate this in the US. You do not care whether or not this is a franchise. It had "Sheraton" on it. You do not care, whether is was in Italy. It was a US brand and you are a US citizen. Keep the case in the US as much as possible!