Legal Question in Family Law in Washington

The short answer to your question is: Yes.

If the father wants to stop that move or do anything about the situation, he needs to bring a paternity action to establish himself as the father, which will lead to entry of a Parenting Plan (a/k/a/ custody orders) and an order of child support. Otherwise, if there is no legally adjudicated father, then no one, including the actual father has legal standing to challenge the mother's intended move.