If I play Shuriken Flip, targeting an enemy unit to deal 2 damage at a battlefield move my unit into that battlefield as part of the spell's effect, and afterward play an Akali's Legend or a move reaction spell to move my unit out of that battlefield, does the enemy unit heal?

Yes, the enemy unit heals.

Once Shuriken Flip resolves and your unit moves into the occupied battlefield, a Combat Showdown is initiated. That combat runs through all its steps, including Combat Cleanup (Rule 461.1.a.1), which heals all units in play — regardless of whether you move your unit out before Combat Damage happens. Per FAQ #8342: "Units heal after combat as long as a combat was started, even if combat damage is skipped or a unit is moved away before damage is dealt."

So the 2 damage from Shuriken Flip will be marked on the enemy unit during the showdown, then heal off when Combat Cleanup occurs at the end of that combat.