The only two questions that they are allowed to ask are the dates of your employment and whether or not you are eligible for rehire. Anything else opens them up for a MAJOR lawsuit in your favor because literally every other question to ask is against the law to ask.
Besides, you can explain to a new employer, "I left abruptly because their gross safety negligence was going to get me or a coworker killed" or "I left abruptly because I received a much better job offer from the company that I just left and that company gave me less than 2 weeks to consider the offer but I needed the pay increase to support my family" blah blah blah.
Your future employer isn't a machine. They're human beings who can make value judgments.
First off, this is pervasive and potentially damaging myth that employers are only allowed to ask about certain basic things like dates of employment or what not. With few exceptions (mostly by state law), a potential employer can ask just about anything they want, and the reference can answer any question -
truthfully. Certain obvious things are off limit such as questions about race/sex/religion/sexual preference, basically all the same things that employers aren't allowed to ask on a job application. But questions about performance, discipline, terms of departure, etc are very much allowed, and you only have a legal recourse if the reference lies about this information.
Now the reason people often cite this myth if that it is sort of a half truth. Many companies, larger ones especially, have very strict policies about what their HR people are allowed to give answers to, and some might even have policies about what they can ask. This is
company policy, not state/federal law. Its done of course to protect the company, since the possibility of a law suit is there if the HR person gives out false information (whether intentionally or not), and they aren't getting anything out of sharing negative information with another company anyways - why take the risk?
But that is the extent of it, and you can and should assume that much more than your dates of employment are going to be disclosed. You should most
definitely assume that information about your termination will be disclosed, particularly who initiated it and what the reason was. I'm inclined to think 'did they give a two week notice' isn't likely to be an oft asked question, or at least not one of much importance, but its certainly on the table.
With that out of the way - what makes you think you'll even be given an opportunity to respond? Your prospective employer doesn't have to give you a reason for why they didn't hire you, and even if they do give you an answer they can just make something up -
which they would most certainly do if its information they shouldn't have obtained in the first place. If whatever they heard from your previous employer is enough to make them think twice about hiring you, you've probably already been passed up for another candidate.