Do you have a source for that? He just made a personal decision that some things in his life were far more important than getting drunk.
This part of it I have heard, but I don't believe the part about Laura threatening to leave. I think that she likely wanted him to quit, but nothing I have read leads me to believe that his drinking was ever to the stage that she was leaving if he didn't quit.
My personal experience If You Suspect You Might Have A Drinking Problem (An Open Letter) is that you must quit for yourself, and that you cannot do it for others. Of course a fear of loss of something important could be a trigger to make one want to quit, but that realization can be arrived at without actual spoken threats.
I don't have an actual source from that -- I thought it was something that was fairly commonly understood (I seem to recall reading or hearing about it during the 2000 campaign).