So how does the new firm know that you were "unprofessional" in the timing of your departure?
That's an easy one. They ask if you're eligible to be rehired.
If you've done something bad or pissed off your management, the answer will be "no".
Since the company isn't disclosing any personal information or making any allegations regarding your conduct, there is nothing they can be sued over.
This is a fairly standard practice in corporate HR.
If you worked for a large business, you could probably dream up a not-entirely-terrible explanation since you know they will not provide any other details. It will still be a mark against, but you can mitigate it quite a bit.
I'm sure it varies by company, but you usually have to do more than just piss off management to be ineligible for rehire. Usually it's things like quitting without notice, stealing, etc.