Let C be the category of homogeneous G-manifolds; the hom sets have a natural topology so you can consider C as an infinity-category. The equivariant homotopy category is the category of contravariant functors from C to the infinity-category of spaces. You build such a functor out of an honest G-space by restricting Hom_G(-,X) to C.

I think this answer is sort of disappointing: it says that all that algebraic topology can see in a G-space are the fixed point sets with respect to subgroups. What are the theorems along these lines that justify this definition?

Let C be the category of homogeneous G-manifolds; the hom sets have a natural topology so you can consider C as an infinity-category. The equivariant homotopy category is the category of contravariant functors from C to the infinity-category of spaces. You build such a functor out of an honest G-space by restricting Hom_G(-,X) to C.

I think this answer is sort of disappointing: it says that all that algebraic topology can see in a G-space are the fixed point sets with respect to subgroups. What are the theorems along these lines that justify this definition?

According to the discussion below, the answer is Whitehead's theorem: any weak G-homotopy equivalence between tame enough G-spaces--at least, all G-CW complexes (Whitehead) and all smooth G-manifolds (Illman)--is a strong G-homotopy equivalence. "Weak" means that the map induces an isomorphism on homotopy groups of all fixed-point sets, and "strong" means that there's an equivariant map backwards so that the compositions are equivariantly homotopic to the identity maps.