When moving onto an uncontested battlefield for a non-combat showdown, can my opponent play an action?

Yes, your opponent can play an action.

According to RiftJudge FAQ #6170 and Rule 344, moving a unit to an uncontested battlefield opens a Showdown. During a Showdown, players have the opportunity to play both [Action] and [Reaction] cards.

Here is how the timing works:

1. Showdown Opens: When you move your unit to the empty/uncontrolled battlefield, a Showdown begins.
2. Priority: The attacking player (you) receives the first opportunity to play an [Action] or [Reaction] to start a chain.
3. Passing Focus: If you pass, focus passes to your opponent (the defender). They may then play an [Action] or [Reaction] to start a chain.
4. Alternating: Focus continues to alternate between you and your opponent until both players pass consecutively without starting a new chain.

As noted in RiftJudge FAQ #3245, players always have an opportunity to play an action spell in a showdown they are part of, allowing them to respond to opponent action spells.