First, you should be clear that a workers' compensation CLAIM is not a LAW SUIT. If anyone asks if you've filed a law suit, the comp claim does not count and the answer (on that basis) would be "no."
Second, it is unlikely that your prior claim would show up on any background search. An employer is not allowed to ask you if you've ever had a prior comp claim. There are ways for them to find out about it but I have never heard of an employer going to those lengths to find out.
Third, if an employer wants to withdraw a job offer, they're going to find a legal reason for doing so and there's probably not much you can do about it. If it does happen, consult a labor/employment law attorney right away.