When I play Riptide Rex, if it doesn't kill the unit, does the damage heal before I move another one of my units?

No, the unit will not heal before you move another unit to the battlefield.

According to RiftJudge FAQ #6441, units are only healed at the end of a combat showdown (after combat damage is dealt) or at the end of the turn. Dealing damage with Riptide Rex's "When you play me" triggered ability outside of combat does not initiate a healing step.

Sequence:

1. Play Riptide Rex: You play Riptide Rex, pay its cost, and it enters the battlefield.
2. Trigger Ability: Riptide Rex's ability ("When you play me, deal 6 to an enemy unit at a battlefield") is placed on the chain.
3. Resolve Ability: The chain resolves, and 6 damage is dealt to the target unit. This damage is marked on the unit.
4. No Healing: No healing occurs at this time.
5. Move Unit: You move another of your units to the battlefield.
6. Combat Begins: If the battlefield is now occupied by opposing units, a combat showdown begins. The target unit will still have the 6 damage marked on it when combat damage is dealt.

Healing will only occur after the combat showdown concludes (as part of the combat cleanup) or at the very end of the turn.