Can you use an action spell after somebody uses an action spell, when your unit is the only unit on the battlefield?

No, you cannot.

According to RiftJudge FAQ #8963, you cannot play an Action spell in response to an opponent's Action spell during a showdown.

Why this is the case:

1. Open vs. Closed State: An Action spell can only be played when the game is in an Open State (the chain is empty).
2. Chain Formation: When your opponent plays their Action spell, it is placed on the chain, and the game state immediately becomes Closed.
3. Reaction Timing: While the chain is Closed, only Reaction spells or abilities can be played. Because an Action spell requires an Open State, it cannot be played until the current chain has fully resolved and the state has returned to Open.

The Sequence:

• Opponent plays an Action spell: The spell starts a chain and the state becomes Closed.
• Your opportunity: Because the state is now Closed, you can only play a Reaction spell.
• Resolution: Once the chain resolves and becomes empty, the state returns to Open.
• Next Opportunity: After the chain is empty and the state is Open, the player with focus may then play a new Action spell.

This interpretation is based on the provided FAQ #8963 and the general mechanics of chain interactions in Riftbound.