It really depends. Bill W., one of the co-founders of AA, struggled with crippling depression off and on his whole life. Depression can be a by-product of alcoholism (especially if it is untreated--meaning no recovery program, no work to relieve the burdens of the past and the stresses in the present and the worries about the future), or it can be entirely independent of it--affecting the sufferer despite a good recovery program--usually due to a chemical imbalance, which can sometimes be relieved by medication.
Is your husband in AA? Unfortunately, there are some well-meaning but badly-informed members who will claim that ALL depression is due to failure to work a "good enough" program. I think Bill W.'s depression should be enough to put that idea to rest. There are also those who will tell other members not to take ANY psychiatric medication. That's wrong, too. Proper medication is a matter for a doctor-patient decision and no one else's. You always want to be sure the doctor does have a good understanding of addiction so that appropriate medication is prescribed, and that its use is monitored.
I hope he is able to find some relief. I have good friends who are happily sober ONLY thanks to their medication AND a good recovery program.