The man also attempts to trivialize the abortion by making it sound simple, but abortion in Spain was not legal at the time. It wasn't made legal in Spain until 1985, and sometimes women died. Many doctors were not willing to risk their careers on being caught performing illegal abortions. Many times, the only doctors who performed them were doctors who weren't very good or whose offices were less than sanitary.
The man wants Jig to have the abortion, but she looks at the situation realistically. The man might leave her anyway, and she might be physically and psychologically scarred from the abortion, even if nothing goes wrong.
The subject of the abortion isn't specifically named, but is alluded to. An "elephant in the room" is when you have a situation or topic of major importance that no one wants to talk about, but the topic is there nonetheless. It can't be ignored, but no one wants to deal with it because they don't know how. Jig tries to deal with it, but the man refuses to. He just wants her to have it, so his responsibility is ended. This is the allusion that leads to the title. The "elephant in the room" is the abortion.