Schmidt, who won bronze at the Tokyo Games, finished out of the medals after a tumble during his routine at the Bercy Arena in Paris.
"I knew I needed to push hard to get onto that podium - those boys are incredible athletes and posted some big scores. So I went out there and pushed hard, and it didn't come off."
Schmidt went into the medal round in fourth place, but went off-line during a twist jump and fell off the trampoline.
He was unharmed, but looked disconsolate as he waited for the score; Schmidt's 19.5 points placed him eighth overall.
The medal winners were from AIN and China.
"It's pretty hard to explain what went wrong, it all happened so quickly and sometimes it doesn't even feel like anything went wrong and you're on the bloody floor.
"It's obviously devastating. It's been ... such an amazing build-up. I'd been training really, really well and the body was feeling good. I'm just disappointed I couldn't deliver on the day."
"I've got more in me; I know I could do better. It's super frustrating, but hey, that's sport. The margins are so small and yeah, I had to give it everything to get on the podium, and I came up short."
Earlier, Maddie Davidson - Schmidt's partner and the other member of New Zealand's trampoline team - put together a great showing in Paris, finishing seventh in the women's final. Team GB's Bryony Page won gold.