Why is the ground wire (connected from battery to engine case) on my gs550 1982 melting?

The connectors on both ends are internally corroded and resistive. Copper conducts heat exceptionally well, so it's probably not the whole wire, but because battery cables are usually almost-too-short, ya gotta replace the entire cable with one as big or bigger, and clean up the connections to the bike and battery. Preventative maintenance could have caught it sooner. While you're at it, replace the battery cable to the starter, too. If one is bad, the other one is on its way out.

It could be a bad connection at the engine or battery or the wire size is wrong for the load. It is not the polarity.

If it's the original, I would also go with bad connections or corrosion at the terminals (where the wire is exposed slightly). Once corrosion starts, it eats under the insulation. Pulling lots of current (ie: running the starter motor a long time) also creates heat. High current through resistance = lots of heat.

You have a ground fault - a short or crimped wire. Please check all wires, and before replacing the melted wire. In the interim, keep the battery disconnected.

this is due to incorrect ground wire cable size and perhaps the reverse polarity ... you are at the point where you need a well versed mechanic .....

What gauge wire are you using?