My unit is in a battlefield and has damage marked on it. If I start a non-combat showdown in the other battlefield will my unit with damage heal?

No, your damaged unit will not heal.

Healing only happens at two specific times: during a Combat Cleanup (which occurs as part of a combat's Resolution Step) and at the end of each turn (Rule 143.3.b, 461.1). A non-combat showdown — moving to an empty/uncontrolled battlefield — does not go through the Steps of Combat, so there is no Combat Cleanup and no "Heal all Units" instruction (FAQ #7796, #9016).

The location of the damaged unit doesn't matter here. Even if it's sitting at a different battlefield from the showdown, there is simply no healing event occurring. It will stay damaged until either a combat finishes somewhere (triggering a Combat Cleanup that heals all units everywhere on the board, FAQ #6111) or the turn ends.