Does a non-combat Showdown heal units?

No. Units do not heal after a non-combat showdown.

Healing only occurs at two specific times: during Combat Cleanup (after a combat showdown — one where units controlled by different players are present) and at the end of each player's turn (FAQ #9016, #9726, #11213). A non-combat showdown — when a battlefield is contested but only one player has units there — does not meet the criteria for combat, so no Combat Cleanup healing triggers.