Sorry, but no... Diverticulitis is essentially a local infection where bacteria get trapped and fester inside a pocket (the diverticulum) of the gut (diverticuli can occur in any portion of the GI tract, but diverticulitis is most often described in the colon). This can lead to abscess, perforation, fistula formation, bleeding. Treatment is antibiotics (surgery if recurrent). Aloe vera is pretty useless here.
No. Diverticulitis involves acute inflammation, infection, and micro perforation of the bowel. Aloe vera juice is a stimulant laxative-not a good thing at that time. Diverticulosis is presence of thin walled pockets in the wall of the colon; unknown if taking aloe vera juice has any beneficial effect, i.e. Preventing complications of diverticulitis or bleeding.