Both Uses

delicacy
in
The Scarlet Letter
(Edited)

It was not an age of delicacy; and her position, although she understood it well, and was in little danger of forgetting it, was often brought before her vivid self-perception, like a new anguish, by the rudest touch upon the tenderest spot.
p. 79.1 *

delicacy = gentleness
And the shame!--the indelicacy!--the horrible ugliness of this exposure of a sick and guilty heart to the very eye that would gloat over it!
p. 181.1

indelicacy = coarseness (not of gentle standards)

standard prefix: The prefix "in-" in indelicacy means not and reverses the meaning of delicacy. This is the same pattern you see in words like invisible, incomplete, and insecure.

Definitions:

(1) (delicacy as in: eat the delicacy)
something that is rare or expensive -- usually a prized food
(2) (delicacy as in: discuss with delicacy)
care and gentleness -- especially speaking or acting with sensitivity and tact
(3) (delicacy as in: offend her delicacy)
the quality of being easily hurt or damaged

(often referring to the fragility of someone's emotional well being when it is easily distressed by something that is offensive or disturbing)
(4) (delicacy as in: delicacy of the brushwork)
pleasant subtlety or fineness
(5) (meaning too rare to warrant focus)