Indelicacy is the noun: the quality of being indelicate, or one act that is indelicate. A remark, a joke, a question that steps on a tender place. Talking about someone's salary at dinner. Asking why a couple have no children, in front of the table. Making a joke that lands as cruel. The word is about manners and other people's feelings, not about breaking the law. It is polite and a little old-fashioned. A novel or a careful adult might speak of "the indelicacy of the remark"; a younger speaker would more likely say that was rude, or you shouldn't have said that. Though actually, indelicacy is often quieter than open rudeness: the person may not have meant harm. They just did not take care.