Can this be fixed at home?

I can't see how the pinion itself has backed out of the FRONT of the gear carrier. The front side of the pinion rides on a tapered roller bearing that is in the back side of the carier; the pinion is trapped between the ring gear and this bearing. It suspect more likely that the front nut that holds the yoke on the pinion is loose and the opposite bearing on the front side of the carrier has loosened up and the pinion is now cocked a bit, and draggin/jamming up the ring gear and carrier.

I would grab the pinion/yoke by hand and gently try rotating it back and forth to see if you can get it to loosen bit and then see if you can straigthen it up by gently, gradualy tigthening the front nut while rocking the pinion back and forth to see if it will pull in the front bearing and strighten out the pinion shaft. That may free up the ring gear and carrier.

Unforturnately, if the pinion has cocked, it is hard to say what has jammed and cracked. But with the pinion partly dragging on the ring gear, then as you rotated the axle by hand the side gear, the sure-grip action 'gripped' on that side (the reason it got harder to turn) and the splines in the diff are now slightly misalgned and gripping the axle splines.

It might also help to just see if you can rotate the axle in the opposite direction. After that, try the slide hammer routine.