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Tennessee is one of the states that offers "protective custody" for individuals that are intoxicated in public, but not causing any disturbance or harm to others.
What this means is that the individual is detained and placed in protective custody for health reasons, but there is no crime attached to this and no arrest record generated.
Based on your description, it is likely that this is what happened to you.
As you cannot recall the specifics of your Saturday evening, you may want to go to the Police Department and ask if they generated a record of any sort (you can contact their records division for this during business hours).
For more information on Tennessee's protective custody system, see: http://www.criminaldefenselawyer.com/resources/criminal-defense/misdemeanor-offense/tennessee-public-intoxication-laws-drunk-pub
The purpose of the protective custody is to ensure that there isn't a detention (and most important for your purposes, no arrest record).
You can call the Nashville PD and ask over the phone (they should help you out without having to actually visit the department in person).
If the PD didn't contact your chain of command, then the Army isn't going to be actively searching it out.
The arrest will still appear, and this may come up when your background checks go through (10 years or 5 years depending on your clearance levels).
Depending on your chain of command, the way they deal with this can be across the board.
(Anecdotally - the advice I have heard is to "keep your mouth shut and sin no more." If the issue comes up and you are unsure of your chain of command - lawyer up, it can be serious)