Q. I interviewed a candidate and referred her to meet the hiring manager, but she was not ultimately hired. Is it appropriate for me to reach out to her (now that we have no business relationship) and ask her out on a date?
A. No. If she expressed interest at any point in establishing a relationship with you that would be one thing.
You are not in a position to ask her out on a date because you only know her due to your position as an interviewer.
She did not create a relationship with you because she saw you and wanted to talk to you, or because you were introduced socially, or for any other reason of her choosing.
Not to be harsh, but she was forced to spend that hour with you, interviewing for a job she didn’t get it. That is not a way to meet women. Leave her alone.