Mediterranean literally translates to "middle of the earth"

Medi = middle Terran = earth Ean = of the

The complete sea name in a native romance language, not English, is Mar de Mediterranean. The word is obviously Latin in origin and the sea has been known as such since antiquity, probably predating Rome (who at the height of their control called the Mediterranean a Roman lake, since they encircled it on all sides. How quaint of them, yea?). This would have been known to Tolkien as the educated, well everybody really, back in the day, had more exposure to Latin, which had spread out widely due to the Church. Martin Luther and Guggenheim started that unraveling, tho Latin is still spoken in many many Catholic churches today.

Midgard does translate to middle earth as well. Tolkien was prob aware of this as well, tho i think that this is lesser well known than Mediterranean, just by the global reach of Spanish.