"a shop where pizzas are made, sold, or eaten," by 1928 in restaurant names around New York city, from pizza with ending as in cafeteria. In 1920s and '30s it sometimes also meant "pizza pie" itself.
Entries linking to pizzeria
pizza n.
"a savoury dish of Italian origin, consisting of a base of dough, spread with a selection of such ingredients as olives, tomatoes, cheese, anchovies, etc., and baked in a very hot oven" [OED], 1931, from Italian pizza, originally "cake, tart, pie," a name of uncertain origin. The 1907 "Vocabolario Etimologico della Lingua Italiana" reports it is said to be from dialectal pinza "clamp" (from Latin pinsere "to pound, stamp"). Klein suggests a connection with Medieval Greek pitta "cake, pie" (see pita). Watkins says it is (perhaps via Langobardic) from a Germanic source akin to Old High German bizzo, pizzo "bite, morsel," from Proto-Germanic *biton- (see bit (n.1)). Ayto ["Diner's Dictionary"] seems inclined toward this explanation, too.
The notion of taking a flat piece of bread dough and baking it with a savoury topping is a widespread one and of long standing — the Armenians claim to have invented it, and certainly it was known to the ancient Greeks and Romans — but it is Italy, and particularly Naples, that has given its version of the dish to the world. ... Since then it has undergone a series of metamorphoses in base, topping, and general character that would make it hard for Neapolitans to recognize as their own, but which have transformed it into a key item on the international fast-food menu. [Ayto]
A pizza is manufactured, as far as I can ascertain, by garnishing a slab of reinforced asphalt paving with mucilage, whale-blubber and the skeletons of small fishes, baking same to the consistency of a rubber heel, and serving piping-hot with a dressing of molten lava. ["Simon Stylites," in The Bergen Evening Record, May 15, 1931]
cafeteria n.
1839, "cafe," American English, from Mexican Spanish cafeteria "coffee store," from café "coffee" (see coffee) + Spanish -tería "place where something is done" (usually business). The sense shifted by 1890s to "self-service dining establishment." The ending came to be understood popularly as meaning "help-yourself" and was extended to new formation with that sense from c. 1923.
Examples of the thing itself date to 1885, but they seem to have become established first in Chicago in the early 1890s by social and philanthropic organizations (such as the YWCA) to offer working girls affordable, fast, light meals in a congenial atmosphere. Their popularity waned after c. 1926, eclipsed by coffee shops, lunch counters, and sandwich shops. Industrial plants began to add them in 1915; schools and colleges followed.