The title of this post is very much an example of availability bias. I've been reading about fallacies, and so now I have the urge to write about things which I can label with the word 'fallacy.'
Also a little bit about definition at the bottom.
The fallacy of emergence is this; the parts do not need the whole to be explained, but the whole needs the parts to be explained. Otherwise, dualism. Put another way, you can entirely describe any situation in terms of its elementary particles and their properties, which means adding a description of the emergent properties is unnecessary, thus extraneous, and thus must be an empty set. I've found that my definition of life* has a flaw; it finds that the species of Earth, in aggregate, is a living being. (For instance, it defends the goal of diversity.)
*(Living things can be assigned goals. I fixed this in my first article about Gaia, written after this one.)
However, Gaia requires the actions of the individual organisms to be explained. But, the actions of the individual organisms do not require Gaia to be explained. The concept of Gaia is a fallacy of emergence, and is completely optional, and thus arbitrary, and thus not real. And indeed, we cannot ever interact with Gaia directly, only with her elements. Problem being that life itself is also emergent.
And I can show that the only way for Gaia not be optional, and thus not real, would be for a Gaia situation to bestow properties. But the properties of Gaia supervene on the properties of the organisms;* Gaia would bestow properties upon Her organisms that cannot be predicted from their components.** This is a literal implementation of dualism, where Gaia forms a new fundamental substance with separate fundamental laws or properties. Logically this is actually quite possible; because of the infinite regression fallacy, fundamental layers are assumptions. If I were constructing a universe, I could easily assume new fundamentals made out of old fundamentals. However, we don't observe this in our universe.
*(If this is not the case, it is not a fallacy of emergence because it is not emergence; Gaia would just be some other, separate thing.)
**(This is more easily seen in a different concrete, electrons. If electrons could form emergent properties, they would in certain arrangements jag left when you expect them to jag right, and it would be impossible to reconcile this with their individual properties and their local space; you would have to take into account essentially incantation-like arrangements of electrons. Imagine a pentagon of electrons which has emergent properties. Usually the electrons would repel each other and fly apart, but the emergent properties would hold the pentagon together as it travelled through space as a particle in its own right.)
None of this is to say that the concept of emergence is useless. Quite the contrary. However, when using it, you should be careful to remember that emergent behaviours have no inherent existence and are merely a different way of looking at the underlying elements.
We can certainly define Gaia and indeed many similar concepts, and find that the components that make them up duly exist, approximately. We can even define properties that Gaia will have, and find interactions that carry out those properties, approximately. However, we need do neither, and indeed if you perturb those definitions slightly in just the right way, they'll still hold. This process is repeatable, and repeatable indefinitely, forming any definition of an emergent property from a perturbation of any other one. They have no objective standard and thus no objective existence. Or, put another way, you can easily find a contradictory definition to Gaia. (Which would require a conjugate definition to fully cover the situation.) For example, bowls. You can define bowls by shape. Also, you can define bowls by function, the function of holding liquid or granular solids. It cannot be both, because an upside down bowl does not have the function, whereas the concave bowl shape is not the only one that holds liquids. The definitions are contradictory. Because both are valid, neither is objectively true.
This is why my only criteria for a sound philosophical inference is that the premises can be found in reality. I don't require any particular definitions, and especially not the accepted definitions. Instead, I suggest you use any definition that is convenient, and simply make sure to note which definitions you're using. Also, as a philosopher, it is not your job to confirm that the definitions in fact match reality, any more than it is the mathematician's job to ensure that theirs match reality.
Certainly, generally speaking I don't see much point in pondering the workings of things that don't exist, but once you get in the ballpark, finding the logical relationships is often as generally useful as math is; while you may not find the definitions where you first thought, it probably can be found elsewhere, or perhaps a small modification effort can turn pure fancy into a cogent understanding.