It ain't true. Not only does the Navy disclaim it, the anecdote shows up in a 1992 collection of jokes and tall tales. Worse, it appears in Stephen Covey's 1989 The Seven Habits of Highly Effective People, and he got it from a 1987 issue of Proceedings, a publication of the U.S. Naval Institute.
It's likely far older than that, because another reader mentioned he saw it passed around as a photocopied joke in the late 1960s while serving aboard either the USS Dixie or USS Truxtun. That certainly agrees with the opinion of Navy sources (as quoted in the news article later on this page); they place the story as being thirty or forty years old.