Unless I'm missing something, the Gompertz function you refer to has an implicit model that the organism's population stabilizes because of Malthusian constraints, a lack of some kind of resource.

But birth rates have been declining in first-world countries for a long time in the absence of this kind of constraint, and the Foreign Affairs article specifically notes that both Malthus and Paul Ehrlich's "Population Bomb" doomsday resource-shortage scenarios didn't come to pass.

The Earth certainly does have some kind of carrying capacity for humans, but that's not the limit that's being it, it's something else.