Argentinian history teacher here. What you have to keep in mind is that while certainly massive, italian migration to the country was not a sudden event, but rather a long process that lasted more than a century (starting roughly at the mid XIX century up until the 1950s). The italian migrants that gradually arrived to the country were surrounded by the local population and had to deal with local authorities, which spoke spanish, and mostly adapted to local practices.

Moreover, italians themselves more often than not didn't even speak what we'd identify as italian, but rather many diverse regional dialects (piedmontese, venetian, genoese, neapolitan, etc.). Add to that the mass of inmigrants from other countries (spanish, germans, syrio-lebanese, polish, etc.), and the strong emphasis placed by the argentine national goverment on mass public schooling and concription (very openly used as tools to integrate the diverse population to the nascent "argentinian" national identity), and then it isn't as surprising that spanish remained the dominant language in the country, being both the official language of goverment and the de facto lingua franca between the diverse groups present at the country.

However, italian did leave a strong imprint in the local dialect of spanish spoken to this day. First, it's influence (along that of many other inmigrant languages) gave rise to various local slangs (the most notable called "lunfardo"), which were widespread amongs the lower classes specially in the pampean region (where most of the inmigrants settled). Eventually many lunfardo words made its way to everyday usage, where they remain to this day. This would directly impact the local version of spanish, which is known as rioplatense spanish, and is spoken widely in Uruguay as well. It has a very distinct sound, quite different from peninsular spanish for example, and it incorporates many loanwords and expressions from italian. A few examples:

*laburo, laburar (work, to work), from the italian lavoro, lavorare

*pibe (boy, young man), from the italian pivello

*mina (chick, broad), from the italian femmina

*fiaca (laziness), from the italian fiacca

*birra (beer), pretty self explanatory

Not only did many italian loanwords made it into everyday usage, but the the intonation of rioplatense spanish has been described by researchers as more colsely ressembling that of the southern italian dialects (most notable neapolitan) rather than any spanish dialect (Colantoni and Gurlekian 2004). On a purely anecdotal level, other spanish speakers (from Spain and Latin America) often describe rioplatense spanish as sounding "como un italiano hablando español" ("like an italian speaking spanish").

SO, to sum it all up, the lack of a unified italian language among the massive italian migrant population along with strong state action prevented italian from de facto becoming a second language in Argentina. However, italian dialects left a strong imprint in the local variant of spanish, which is still going strong to this day.

Hope this helps!

COLANTONI, L., & GURLEKIAN, J. (2004). Convergence and intonation: Historical evidence from Buenos Aires Spanish. Bilingualism: Language and Cognition, 7(2), 107-119. doi:10.1017/S1366728904001488

*Small edit to fix some spelling