I move in against a Tide Turner with an Empowered Akali deadly weapon. I assign the 2 damage to the Tide Turner. Can my opponent play Star Crossed on both units and save his Tide Turner?

Yes — if that 2 damage is Akali's Empowered move trigger, your opponent can play Star-Crossed in response and save the Tideturner. Star-Crossed is a [Reaction], so it's playable while your trigger is on the chain. It resolves first (LIFO), returning both units to hand; when your trigger then resolves, Tideturner is in a non-board zone and is no longer a legal target, so the damage instruction is ignored (Rule 359.3.e.5).

One timing caveat: if you mean combat damage has already been assigned — no. Once the Showdown closes and assignment begins, there's no window to play anything (FAQ #10832). Your opponent's last chance was to play Star-Crossed before the Showdown closed, which would also save the Tideturner by removing both units before the damage step.