For a Navy officer, the only way a Bad Conduct or Dishonorable discharge is awarded is for the officer to have been convicted at a General Court Martial (which is convened by a Flag or General Officer, much like you saw in "A Few Good Men").
A conviction at a General Court Martial carries the weight of, and is in fact, a felony conviction. I doubt that Kerry was walking around with that until 1978 or 2001, as he would've been found out long before now.
Like I said, it's not inconceivable that he could've been awarded a General discharge, and that it was mailed to him upon fulfillment of his Individual Ready Reserve commitment, and he never got around to having it upgraded by the Board of Corrections of Naval Records (BCNR) until much later in his budding career.
What would a general discharge entail? It doesn't sound too negative.