In "Hills Like White Elephants," Hemingway puts the reader in the position of a voyeur and an eavesdropper. The reader can only obtain information from what is done and said by the two principal characters. Evidently the decision to have an abortion has already been made. They are waiting at a transfer station for a train that will take them to Madrid. The girl obviously does not want to go through with the "operation," but she has reluctantly agreed to do so. The man keeps talking to her because he is afraid she will change her mind.
It is highly unlikely that the girl would use the word "abortion" because it is obviously something she doesn't want to think about or talk about. So the only person who might use that word would be the man called "the American," and he doesn't want to use the word because (1) he doesn't want to remind the girl of what is in store for her, and (2) he keeps trying to assure her that it is something very simple, not really an operation at all. In other words, he is denying that this is an abortion but trying to portray it as an "induced miscarriage."
Besides that, they have to be discreet. This was the 1920s. Abortions were illegal in most countries--and they were probably even more seriously punished in a Catholic country like Spain. He doesn't know what kind of trouble they could get into if things didn't work out perfectly. He probably only has a name and an address. He is trying to appear more calm and confident than he really feels. In a worst-case scenario, the girl could die and he could go to prison for life--or even be executed!
Even though they are speaking English, the word "abortion" might be understood by someone who overheard it. The girl and the American are conspicuous because they are foreigners speaking in a foreign language. The reader can understand what is going on without hearing the word "abortion"; why shouldn't others be able to figure out what is going on between them just from their body language and the other clues such as facial expressions and tones of voice?
It would be hard to think of a sentence in which the American would use the word "abortion." Would he say something like: "When we get to Madrid, I'll go out and make all the arrangements for the abortion?" It just doesn't seem plausible that he would want to air that awful word. He has had a hard enough time talking Jig into having the "simple operation" and he is terribly afraid she will change her mind even now. Why call a spade a spade?
Part of the appeal of this minimalistic story is that the reader has to try to understand what is happening by deduction from what can be seen and heard. There are many questions left unanswered besides the question of why the word "abortion" isn't used. Who are these people, anyway? Where are they coming from? Are they married? How are they paying for all their traveling around? What finally becomes of them? Does the girl change or mind? Or does she go ahead with the procedure they don't dare to name aloud? What will their relationship be like afterward?