Do units heal after non-combat Showdown?

No, units do not heal after a non-combat showdown.

According to RiftJudge FAQ #9016 and #9220, healing only occurs after combat showdowns and at the end of the turn. A showdown only qualifies as a "combat" if it is contested and occupied by units controlled by different players. If a showdown does not meet these criteria, it is non-combat, and no healing is triggered.