The incident occurred during Hoffenheim's 3-1 home defeat to Borussia Monchengladbach on Saturday.
Kerem Demirbay had given the hosts the lead after 25 minutes, but a poor second half saw Hoffenheim slip to their second defeat in a week, after losing 1-0 to Werder Bremen in the DFB Pokal on Wednesday.
Nagelsmann told Bild: "This was a stupid action from me. I wanted to throw into the wall, but the bottle neck was stuck in my sweater. That cannot and will not happen [again].
"I really wanted to throw the bottle from the coaching zone into the wall. Thank goodness it did not fall directly onto the fan, but I'm sorry.
"It was not my intention to throw the bottle into the audience."
Nagelsmann apologised to the supporter in person immediately after the game.