The key thing that World War I did was make it socially uncool (besides legally suspect) to use non-English languages for many daily life purposes, and for just about all official purposes. But in fact both of my parents, who attended high school in the 1940s and early 1950s, took foreign language classes in high school. Taking a foreign language was simply considered part of a sound academic high school education in those days, never to be omitted. What my parents missed out on was PRIMARY schooling conducted entirely in the medium of another language, which both of my mother's parents had even though both were born in the United States. (My maternal grandparents attended school in German, in two different Great Plains states.)

The most important factor in making English the dominant language in the United States is that it is the interlanguage (dare I say "lingua franca"?) that unites all ethnic groups here. Only about a quarter of the United States population actually has ancestry from English-speaking places (which, once upon a time, meant the British Isles only). Sure enough, only about one-fourth of my own ancestors were English speakers when they arrived in North America. But all the people who arrived from other places to the United States found that English was the language they could count on as they traveled by canal boat or stage coach or railroad or sent postal letters or telegrams and eventually made telephone calls to one another, so ultimately everybody learned English. The United States is remarkably unified by a common variety of English, often the sole language of stubbornly monolingual people, even though Americans come from all over the world.

Edit, as an aside, I'll mention that Esperanto (mentioned in another comment) failed because it is not at all a neutral choice, but is stuck with altogether many weird features of a tiny subset of Indo-European languages that make it quite hard for many learners to learn.

http://www.xibalba.demon.co.uk/jbr/ranto/

They're probably exaggerating the state of language education in most of USA before those laws. However, one could also imagine the thought process of prospective teachers who lived then: "Well my Greek is pretty good, and with a couple of years' more study I could teach at the high school level. On the other hand, we might get in a war with Greece someday, so instead I'm going to teach math."