Distal end of clavicle fracture and slight dislocation of AC joint after falling down from stairs now on sling. Will it heal without surgery?

Varies. Varies somewhat based upon the degree if displacement. Most clavicle fractures do well with nonoperative care, and most are in the middle third of the clavicle. But the one type we do have to fix sometimes is badly displaced lateral clavicle fractures that involves the AC joint. I would discuss with your md.