heal after showdown

It depends on whether the showdown was a combat showdown — those heal all units; non-combat showdowns heal nothing.

If units controlled by different players were both present (a real combat), the Combat Cleanup during the Resolution Step heals all damage off every unit in play at once — including units sitting at Base or at other battlefields that never fought (Rules 466.1.a.1, 461). But a non-combat showdown (you move to a battlefield and no enemy units are there) has no combat damage step and no Combat Cleanup, so damaged units just stay damaged (FAQ #9726, #11213).

Either way, all damage is also cleared for all units at end of turn during the Expiration Step (Rule 317.2.b). Note that healing only clears damage marks — it doesn't remove Might buffs or counters.