I don't mind telling people why I don't drink. I share the information when someone asks (I don't bring it up randomly!!) so that there isn't a stigma. When I first joined the site, I wasn't sure if I wasn't being overly dramatic about my drinking. The more I read the more I realised that yes, I am an alcoholic. I think people should know that there is no certain image or demographic to this condition and that everyone who drinks is at risk. If they don't want to be around me because of it, then good riddance. It usually brings up people justifying their own drinking habits so already it's no longer about me and the conversation just carries on...
I stopped drinking on my own and didn't consult a doctor - I was on day 3 or 4 when I joined SR and read the dangers of detox. Fortunately, I didn't have any withdrawals but I consider myself very lucky. I think that if you don't wish to disclose the information to the doctor, then maybe that's telling you that you don't trust them and should change to someone you are comfortable disclosing the information.