Do conquers on one battlefield cause damaged units on the other battlefield to heal?

No, conquering a battlefield does not cause units on other battlefields to heal.

According to RiftJudge FAQ #6111 and #4324, units heal at the end of combat and at the end of the turn. Conquering a battlefield is a game action (Rule 464.1), but it is not combat. Therefore, it does not trigger the "Combat Cleanup" step where healing occurs.

To summarize:

• Healing occurs automatically at the end of combat (for all units everywhere) and at the end of the turn (for all units everywhere).
• Conquering a battlefield does not trigger a combat cleanup, so no healing occurs as a result of that action.