Why will a radioactive rock never become completely stable?

Technically speaking, depending on the sample, the rock could become “stable.”

The radioactive decay of a radioisotope is expressed as a half-life equation; half-life is the colloquial term that describes how long it will take for half of the radioisotope to decay into another isotope or element. For example (if I remember correctly), Carbon-14 has a half life of 5,780 years. This means that in a 100% sample of C-14, after 5780 years passes, only 50% of that isotope would remain; another 5780 years, and only 25% would remain (half of half). Based on this principle, it seems like a sample could never fully decay because there’s always an amount that smaller than the current amount.

However, if the sample has a very short half life (milliseconds or nanoseconds) the sample would reach nigh-full decay eventually. At this point, it is considered “stable.”