"Nosedive" is one word. It means to make a sudden drop like an aeroplane descending nose first from the sky. It's hard to understand how it works in the sentence you've given. I would have expected: "When she told him to go home his spirits nosedived."

"Nosedive" can mean he wanted to shoot his shot and get her number, maybe he likes her and is persistent. Maybe he did a literal nosedive on her in a comedic sense, like jumping on her headfirst.