mid-13c., "yell (something) out, utter" (transitive); c. 1300, "beg, implore; speak earnestly and loudly; advertise by calling out," from Old French crier, from Vulgar Latin *critare, from Latin quiritare "to wail, shriek" (source of Italian gridare, Old Spanish cridar, Spanish and Portuguese gritar), which is of uncertain origin.
Perhaps it is a variant of quirritare "to squeal like a pig," from *quis, echoic of squealing. Ancient folk etymology explained it as "to call for the help of the Quirites," the Roman constabulary.
The meaning was extended 13c. to the sense "shed tears" that had formerly been in weep, which it largely replaced by 16c., via the notion of "utter a loud, vehement, inarticulate sound." To cry (one's) eyes out "weep inordinately" is by 1704.
Most languages, in common with English, use the general word for "cry out, shout, wail" to also mean "weep, shed tears to express pain or grief." Romance and Slavic, however, use words for this whose ultimate meaning is "beat (the breast)," compare French pleurer, Spanish llorar, both from Latin plorare "cry aloud," but probably originally plodere "beat, clap the hands." Also Italian piangere (cognate with French plaindre "lament, pity") from Latin plangere, originally "beat," but especially of the breast, as a sign of grief. Related: Cried; crying.