I believe both descend from "rap" as a form of rebuke, in one case synonymous with criticism, in the other synonymous with a conviction in a court of law.
I am curious about is the origin of 'rap' as an informal discussion: I want to rap to you today—lyric spoken by Maurice White of "Earth, Wind, and Fire."
I believe that's the origin of the noun Rap as a music genre, which is now mostly known as Hip-Hop.
I'm pretty sure it's jazz talk, so I'd look at urban black culture of the early 20c.
Could have ended up being knock music or call music?
edit: your "rap" is (usually) how you talk to women, and going "calling" was once the same sort of euphemism.
Way to make us feel old! I was at school when children started to have phones, through to MSN and early smart phones, but 'calling on' people was at least the bridge between that and RL interaction. (I'm sure we texted things like 'call on you later'.) Is it not still? Whatever the technology isn't there a need for that?
Obviously language can fade independently of technology, but it was quite funny to note some of the same weird school playground phrases familiar to both me and my father at the same school(s) 30y apart.
OED attests "rap" in this sense of punishment back to the 1860s at least, and "face the rap," "beat the rap," etc., to the early 1900s. "Rap sheet," the 50s.
From Collins dictionary of British English
""" Rap: 2 (verb) in the sense of reprimand
Definition to rebuke or criticize sharply """
Other interpretations would have been too good to be true.