What is the exact ruling for Falling Star damage heal after combat?

Yes — damage from Falling Star is fully healed after any Combat ends, and again at end of turn.

When a Combat Showdown (attacker vs. defender units) reaches its Resolution Step, the Combat Cleanup heals all units in play — every battlefield and Base, whether or not they fought (Rule 461.1.a.1; FAQ #4324). So a unit hit by Falling Star at Base, or at a completely different battlefield, is healed by a combat that happened elsewhere.

Key caveats:

• Non-Combat Showdowns (conquering an empty/uncontested battlefield) have no Combat Cleanup and no healing — moving to an open battlefield does not heal (FAQ #4578).
• Healing also happens in the Ending Special Cleanup (end of turn, Rule 317.2.b), so damaged units recover then regardless.