That depends on what kind of background check the employer did. If the employer used a 3rd party company to obtain the background report then the employer obtained a “consumer report” that is regulated by the federal Fair Credit Reporting Act (FCRA). Those reports must be destroyed after use by the employer and cannot be disclosed to others.
If the employer did the background investigation itself, e.g. by doing internet searches, doing its own search of public records, etc., that is not regulated by federal law or Alabama state law either. In that case, the employer is free to provide that information to the doctor.
The fact that it is public record means that anyone could have gotten that information, so it’s hard to argue that it’s confidential information anyway. If the feds are targeting doctors in Alabama, the doctors may themselves be checking local public records to see if any of their patients have been arrested on drug charges. The last thing a good doctor wants is to lose his/her DEA license to prescribe drugs because the feds suspect the doctor is selling narcotic pain prescriptions to patients who don't need them or who abuse them. So they may well be on the look out for possible drug abusers and cut them off when they find them.

