You were in the service for 3 years- that is a big gaping hole in your application if you do not include it. It is going to look fishy. Bottom line- be honest and a general discharge is not the same as a dishonorable one, right? You can argue honestly that there was an inability to adapt to military life. You might even be able to turn this into a positive somehow.
In any event, you are going to need to disclose this information to get your medical license- if you lie, they will deny you a license. So don't. Also, a lot of physical impairment networks are going to consider arrests to be indicative of a drinking problem even if it is underage? What was the official charge? Public drunkeness, open container? DUI?