No, I would not call them sick because it's an identity.
I think depression is a sickness of spirit, thoughts, and emotions. I would say someone has depression, or is impacted by such a condition.
I wish English were like Spanish. Spanish has two versions of the verb to be -- estar and ser. I would not tell you, "Eres enferma," that would be calling you a sick person. I would say, "Estás enferma," it is conditional and not representative of who you are. Spanish does the same for all conditions -- height, weight, etc.