The differential on '82 Fiat Spider has locked up. Is this thing toast or is there something I can do to fix it?

You'll have more luck getting a reply if you post your query in the 124 Spider forum.

As said above, re-post in the correct forum. Yours is rear wheel drive, the Barchetta is front wheel drive. Different issues of diff locked.

Usual causes of a diff locking are central pin moved and jamming, or bearings breaking up.
If it locked at very low speed, internals are likely to be salvageable, if moving at anything more than walking pace, internal damage very likely.
Whatever, diff will have to come out. Cause is usually obvious once it is out. Then a specialist will advise. If just a loose pin, they can usually be fixed easily. Specialist will of course wish to replace bearings and set up the backlash, which is a good idea, then it'll be good for a long time.