I think this is kind of a broad question, and when you say screams in rock music it's kind of a different definition to how it is used everywhere else. I think of everything from Deep Purple to Iron maiden, opeth and beyond in rock, but in everything else it can have multiple meanings.
When I started searching it seemed that Bel Canto has some early screaming from Queen of the night in the 17th century, but then I found earlier examples of a similar sound in 1600. I believe you could trace it back even further by listening to early vocal music, but there are far better hunters on this site than me for that!
Again it's down to what you define as a scream. If it's a pure resonant tone then you can say that most vocal music was screaming at one point!
I can't give you a complete history on this one, but I can give you some examples in my mind that are outside metal/rock. In chronological order
Cruda Amarilli - Claudio Monteverdi (1600)
This may not fall into the screaming category, only I'm not sure where the line is if we say that queen of the night is screaming, because both it,this and St Matthews Passion have a similar quality
Not All My Torments - Purcell (1659 - 1695)
St Matthews Passion - Bach (1727)
Queen Of the Night - Mozart (1791)
A very large chunk of opera could be considered screaming if it's anything like the above. in that regard screaming could be as far back as the Bel Canto technique
Dies Irae/Requiem - Verdi (1874)
I could go on for a while with that style, it's beautiful, but I'll leave bel Canto for now!
I Put a spell on you - Screamin Jay Hawkins (1956)
Eight Songs for a Mad King - Peter Maxwell Davies(1969)
This makes heavy use of extended vocal techs, especially screaming. It also has an extremely crazy score (while still being playable)
Lovin' You - Minnie Ripperton(1975)
Phantom of he Opera - Andrew Lloyd Webber (1986)
Ok, I said no more Opera, but this one's technically a musical, totally different thing ;)
there's also a nice little compilation you might like on Youtube here
Hope that that helped to answer your question, It was a lot of fun hunting around for all the songs! Almost put Crazy Horses in there, but that was a synth of some sort :(