Is it possible that, one day in the distant future, it might be true that continental drift would stabilize? For the sake of discussion, we could suppose "stability" would mean less than a meter of movement within a thousand-year span relative to surrounding continents, across all continents (or something along those lines - I'm also curious if there's another definition of "stability" that would be more meaningful here).
I say that because it is heavily supposed that Mars, which is now tectonically stable, was active in the past. The reason why it has reached equilibrium is still a matter of discussion, and its tectonic history is merely hypothetical. But right away I'd say that the reason why Earth's tectonic activity could end converges on the reason why Mars, possibly, is stable nowadays: lack of internal energy.
You see, the current geological model that explains the movement of the tectonic plates is based in mantle convection. The mantle is strictly solid, but on a large timespan, i.e., on the geological scale, it behaves as a fluid, which (sort of) means it has enough mobility to move, and therefore for convection to happen. But here's the catch - convection is a form of heat transfer. So we can interpret that convection, and therefore plate tectonics, are but a form of heat to be "expelled" from the inside to the outside. And naturally, at some point energy should end. But at which point? In my opinion, for an astoundingly long time (over 1 billion, and perhaps even 10 billion years). Earth's been spitting energy for almost 4 billion years now (before that, it was actually gathering energy through the kinetic energy from the impact of planetesimals, which can be thought of as huge meteorites) and it still has enough energy to heat the inner core to over 4000 Celsius degrees (that temperature is due to pressure & Earth's remaining energy, though).
It is important to note that maybe due to the fact extreme depth is intrinsically associated with increased temperature, that pressure-driven temperature might suffice to generate mantle convection. But we wouldn't have oceanic crust generation (no inner heat - no partial melting next to mid ocean ridges - no oceanic crust generation), or it would be very limited. And besides, Earth is an open system with the universe, so it loses energy, and eventually it has to run out of it.