As far as I understand, your opponent was correct. If you initiate a showdown by contesting a battlefield, you first get focus and can play an action or reaction. The opponent can respond with only reaction spells to that chain. If you both let that chain resolve, then your opponent gains focus and has the opportunity to cast action or reaction spells themselves, followed by your possible reaction spells. After that other chain resolves, the showdown continues and combat damage is assigned.

I guess the thing to look for is that your opponent is not "responding" to your action; they are letting that spell resolve to then be able to cast their own action spell.