A movie about a young white plantation owner son falls in love with a black slave/
A white plantation owner daughter is jealous of the black slave girl because the white man is in love with her. She has the black girl beaten so bad, she dyes.

This movie aired on CBS in the late 1980's are early 1990's

If it was a TV miniseries, I'd suggest "Beulah Land" (1980) though I don't recall if it aired on CBS. Beulah Land (TV Mini-Series 1980) - IMDb (http://www.imdb.com/title/tt0080197/)

The "North and South" miniseries (two of them) aired in the mid-Eighties, but I don't recall all of the storylines.
North and South (TV Mini-Series 1985) - IMDb (http://www.imdb.com/title/tt0088583/?ref_=sr_1)
North and South, Book II (TV Mini-Series 1986) - IMDb (http://www.imdb.com/title/tt0090490/)
Part III was aired in the Nineties.

I doubt that "Mandingo" (1975) would have aired on network television though it has some of the described elements: son of plantation owner involved with slave girl; his wife also involved with a slave. However, I don't recall whether the wife kills the girl though her husband kills her and her lover. But, this movie is so controversial to this day that I can't imagine it being aired.

I think "Beulah Land" does share some of the story ideas. In histrionics/melodrama, it's somewhere between the "North and South" films and "Mandingo".