Luckily, no. While an alpha male generally has all or most of these symptoms, they could be caused by many different conditions and so having these symptoms alone would not be enoudh to give you that diagnosis.
IMO, one of the traits of "alpha males" is that they identify themselves as such. So if you're not convinced that you're an "alpha male" yourself, you probably aren't one.
Also, ironically, they very often are in a cultlike following of some other (in their mind) "alpha" guy.
think the joke is more that most people with anxiety/depression would fit that incredibly vague characterization.
\ mgreg