The word cutting alone has had a figurative meaning since before the publication of the biblical translation you mentioned, according to Oxford English Dictionary.
That acutely wounds the mind or feelings.
Examples:
1582 R. Stanyhurst tr. Virgil First Foure Bookes Æneis iv. 78 Dido the poore Princesse gauld with such destenye cutting, Crau's mortal passadge.
1652 C. B. Stapylton tr. Herodian Imperiall Hist. xiv. 115 Their cutting quips and wonted jeering.
The phrase "cutting remark" is not an idiom by itself per se, but a use of this figurative term "cutting" applied to a remark. Other terms can be described as cutting, like in the example from 1652, quips.