The 55-year-old actor has been proud of Maya's work as Robin Buckley on the Netflix sci-fi horror hit, which recently ended with its fifth and final season, and he had some advice for her now that the show is over.
Speaking to Variety, he revealed he told her: "Time to move on. You got to do it, and move on. Don't look back."
The father-and-daughter duo worked together on 2023 biopic drama Wildcat - based on the life of author Flannery O'Connor - which Ethan directed.
However, he's keen to act with Maya in the future.
"Heck yeah, are you kidding me? I want to be with the star of Stranger Things!," he said
Having told his daughter to look to the future, Ethan has been taking his own advice.
In 2025 alone, he appeared on the big screen in Blue Moon, She Dances and Black Phone 2, while he had a main role in TV crime comedy drama The Lowdown, as well as voicing General Anthony Wayne in docu-series The American Revolution.
"Why can't I sit still? Why do I work so much? I love it. I am amazed when people do one movie every five years," he said.
"I really like making movies. I started when I was young and fell madly in love with the process. You can't do it alone. It has magic attached to it.
"Once you felt that feeling of making something bigger than yourself and you watch it connect with audiences, I chase it. I'm just chasing."
Hawke recently admitted he always suspected Maya would follow in his footsteps.
Speaking to Sydney Sweeney for Variety's Actors on Actors series in December, he said: "I knew when she was about four that she was gonna be an artist. And I knew that she was gonna be a very good one...
"That was her safe place. Watercolouring, dancing, and singing all throughout her childhood.
"There were a lot of things about her childhood that were really, really hard and complicated, and things I regret for her."
The Black Phone star didn't delve into Maya's early struggles, but recalled a moment when a teacher asked if she was "happy", and she had an answer beyond her years.
"She was about 13. (Maya said), 'I don't think that's a very interesting question," he said.
"'I think there's a lot more interesting questions than whether I'm happy or not. Am I happy? No. But I don't aspire to be happy.'
"So I never worried about her going into the arts, because I knew that was gonna save her life."
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