If I hold a battlefield with one unit and move it back to base, if I move another unit into that same battlefield that is now open does a showdown begin?

Yes. Moving your last unit off leaves the battlefield uncontrolled, so moving a fresh unit into it applies Contested and opens a non-combat showdown during the next cleanup (Rule 344.2, 445).

Important nuance: the showdown only re-establishes your control — you do not score again at that battlefield, since you already scored it via Hold earlier this turn (Rule 447; FAQ #7971). Also, unless your opponent has a unit there when it opens, it's a non-combat showdown, so no Combat Cleanup/healing occurs.