Could prostatitis cause severe pain in my scrotum?

Possibly. Prostatitis can have a wide range of symptomatology. It can range from minimal symptoms to severe discomfort with urination. If there is an acute flare up of symptoms, there can certainly be referred pain to the pelvic area, scrotum, and rectal/anal area. The treatment should be focused on the prostate and as this problem resolves, the other symptoms will usually get better as well.
Scrotal pain. Simple answer is - yes. You should see a urologist who ca properly diagnose and treat yuo as soon as possible to eliminate the pain.