When my opponent goes to my battlefield that I hold and don't play an action spell, can I play it then?

Yes, you can play an Action spell in this situation.

When an opponent moves a unit into a battlefield you control, a Showdown begins. During a Showdown, players have the opportunity to play spells in an alternating fashion.

According to the Showdown Flow and Rule 732:

1. Action is a permissive keyword that allows you to play a spell during a showdown, even when it is not your turn.
2. If your opponent moves into your battlefield and chooses not to play an Action spell, they are essentially passing their initial opportunity to act.
3. Because it is a Showdown, you have the opportunity to play an Action spell as long as you have Focus (the permission to start a chain).

Sequence:

• Opponent moves a unit to your battlefield (Showdown opens).
• Opponent passes their opportunity to play an Action spell.
• You receive Focus and can play an Action spell to start a chain.
• Once you play your spell, the state becomes Closed (a chain exists), and players may then only play Reaction spells until the chain resolves.

As noted in RiftJudge FAQ #2648, you can play Action spells during an open showdown when an enemy moves a unit into a battlefield to contest it.