Do kidneys heal on their own after kidney stones?

Usually if the stone passes. Depends on type of stone and where it is. Many have stones that never cause a problem so they at least must heal on their own.

Recurrent and persistent kidney stones can leave residual renal damage which means chronic kidney disease. Rarely, one impacted ureteral stone can destroy the whole kidney or causes irreversible damage. Fortunately, kidney stone is usually singular and not recurrent so it won't result in residual damage whether it passes out spontaneously or it is taken out by procedure.

most kidney stones do not cause any kidney damage. Any damage to the kidney depends on whether there was obstruction- and for how long and whether there was infection and how severe.