If I have a damaged unit and I move it to an open battlefield and conquer, does my unit heal?

No, your unit does not heal in that scenario.

According to RiftJudge FAQ #7796 and #10974, 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 on the board 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 (Rule 143.3.b).