Do you think memes are something new?
In 1928, a cartoon by Carl Rose, which was captioned by E.B. White (of Stuart Little and Charlotte's Web fame), was published in The New Yorker. Broccoli was a new thing on the American plate, having been introduced by Italian immigrants on the East Coast.
For whatever reason, the phrase caught on: "I say it's spinach" came to mean "to hell with it," and eventually "spinach" came to mean something worthless. Elizabeth Hawes, for example, titled her 1938 autobiographical exposé of the fashion industry, for example, Fashion is Spinach.
Alexander Woolcott used the phrase in 1934's While Rome Burns ("I do not myself so regard it. I say it's spinach.") S.J. Perelman was an American humorist who wrote (among other things) two Marx Brothers movies (Monkey Business and Horse Feathers) and, in 1958, a TV version of Aladdin with music by Cole Porter; he wrote a story in 1944 for the Saturday Evening Post called "Dental or Mental, I Say It’s Spinach."
Speaking of Cole Porter, other musicians used the phrase, too.
As with most immigrants, Israel Isidore Baline (better known as Irving Berlin) felt he needed to be more American (and more patriotic) than anybody around him. (It's pretty common with a person "born-again" into any subculture - religious, societal, or any other coherent group.) His way of doing that was to be more in touch with popular culture than anybody else. So he wrote songs that reflected "the common man" - many of them, we would now consider racist (but that was very common in America at the time).
In 1932, Berlin was already a successful musician, when he wrote the musical Face the Music. (That wasn't redundant. Shut up!) In it, he included the song "I Say It's Spinach (And The Hell With It)."
The lyrics start at 1:14, if you're in a hurry.
Also, despite the impression you get from the video, the first Popeye cartoon was made by Fleischer Studios a year after this song was recorded, in 1933. And at the end of the song, the Popeye-like voice is by a man named Poley McClintock. He'd been using the low, croaky voice on records since 1927; some people have suggested that voice actor William (Billy) Costello based the voice of Popeye on McClintock.
So, even without the internet, a single meme could find a place in the popular culture of America before parts of the country even had running water.