The labels on the luggage "from all the hotels where they had spent nights" suggests to me that this couple has been together for a long time and that they are either married or have been representing themselves as married. I am more and more inclined to believe that they are actually married. The labels actually advertise the fact that they have been living together in hotels, which was a lot more sensitive issue in the 1920s than it is today. Some hotels would actually refuse to rent rooms to unmarried couples. Most European hotels required guests to show their passports and even leave the passports with the desk clerk for a short time while he presumably checked the local police.
The relationship between the man and the "girl" is not casual or temporary. They are carrying a lot of luggage, another indication they have been together a long time. The American tells Jig no less than five or six times that she can have the baby if she really wants to, but there is no mention of marriage by either of them. This also makes me assume they are married already, because Jig certainly wouldn't want to have a baby out of wedlock. If they were not married, the American would say, at least once, "We'll get married and you can have the baby."
Another reason for thinking they are married is that Hemingway often wrote stories that were autobiographical. In the late 1920's he was married to his first wife Hadley and they had a baby they called Bumby. Hemingway was trying to be a freelance writer and was having very little success. They were living in Europe partly because it was much cheaper than the U.S., and Spain was probably the cheapest place of all. It seems entirely possible that when Hadley became pregnant, Hemingway tried unsuccessfully to talk her into having an abortion.
Hemingway said that he often wrote stories to "get rid of things," and he may have written "Hills Like White Elephants" to try to get rid of his guilt feelings about Hadley, from whom he was divorced at about the same time "Hills Like White Elephants" was published.