Does heal happen on Battlefield Showdown if my damaged unit moves to an open battlefield and conquers?

No, healing does not occur in that scenario.

According to RiftJudge FAQ #9649 and FAQ #9131, there is a critical distinction between a combat showdown and an open showdown:

• Combat Showdown: Occurs when a unit moves to an opponent-controlled battlefield. This triggers combat, which includes a "Combat Cleanup" step where all units heal.
• Open Showdown: Occurs when a unit moves to an empty (uncontrolled) battlefield. This is a showdown, but it is not combat. Because it is not combat, there is no combat cleanup, and therefore no healing occurs.

While conquering an empty battlefield allows you to score, it does not trigger the combat cleanup required to heal your units. Units only heal after combat concludes or at the end of the turn.