Interestingly, this practice does appear to go back to the Romans. Wilson D. Wallis, "The Romance and the Tragedy of Sneezing" (1919), writes:
Petronius, Apuleius and Pliny tells us of the Roman custom of saluting one who has sneezed [...] The Roman salutation was "Salve!" equivalent to our "May you have health!"
An indirect form of this, salvere iubet, is used accordingly in the Satyricon:
Dum haec ego iam credenti persuadeo, Giton collectione spiritus plenus ter continuo ita sternutavit, ut grabatum concuteret. Ad quem motum Eumolpus conversus salvere Gitona iubet.
I was just inducing him to believe me, when Giton burst with holding his breath, and all at once sneezed three times so that he shook the bed. Eumolpus turned round at the noise, and said "Good day, Giton."
More recently, in Conversational Latin for Oral Proficiency, John Traupman translates the English "bless you," specifically after a sneeze, as "salvē or salūtem."