Legal terms change from country to country, and I'm not a specialist, so here are some expressions that may work, just from speaking Spanish.
- No tengo (nada) más que decir/añadir/agregar.
- Concluyo/termino mi alegato.
- Está todo dicho.
Someone just told me they've heard:
-He dicho. (Which is used in oratory to conclude a formal speech.)
-Lo dicho, señor Juez.
These two should be understood as "I'm finished talking".
I don't know if any lawyer in Mexico would say "la fiscalía/defensa descansa", but I guess it would be understood in context.
"Abandono el caso": You're right. For me, it means the lawyer hasn't finished their job and they're dumping the client in the middle of the process.
In an online dictionary I found "concluyo mi caso", but "caso", for me, it's the whole process, not only the allegation. So, "concluyo mi caso", means that the accused has been sentenced and the whole process is over.