The usual statement is that if $X\to Y$ is an equivariant map of $G$-CW complexes and if for every closed subgroup $H$ the induced map of fixed-point spaces $X^H\to Y^H$ is a homotopy equivalence then in fact the map has an inverse up to equivariant homotopy.

This is part of the following picture: The category of $G$-spaces has a model structure in which a $G$-map $X\to Y$ is called a weak equivalence (resp. fibration) if and only if for every subgroup $H$ the resulting map $X^H\to Y^G$ is a weak equivalence (resp. fibration). As generating cofibrations you can use the inclusions $G/H\times S^{n-1}\subset G/H\times D^n$ for all $H$ and $n$. In particular $G$-CW complexes are cofibrant. And strong homotopy equivalence in the model category sense becomes equivariant homotopy equivalence in the obvious sense. Thus the statement above becomes an instance of the general principle that for cofibrant objects every weak equivalence is a strong equivalence.

By the way, a reasonable question is, is the analogous statement true with orbits instead of fixed points? That's not true, though. Take a CW space Z that is acyclic but not contractible. Let X be the suspension of Z, and let G of order 2 act on it by switching the two cones, with Z as fixed point set. Then both X and the orbit space are contractible, but X is not equivariantly contractible because the fixed point set is not contractible.

The usual statement is that if $X\to Y$ is an equivariant map of $G$-CW complexes and if for every closed subgroup $H$ the induced map of fixed-point spaces $X^H\to Y^H$ is a homotopy equivalence then in fact the map has an inverse up to equivariant homotopy.

This is part of the following picture: The category of $G$-spaces has a model structure in which a $G$-map $X\to Y$ is called a weak equivalence (resp. fibration) if and only if for every subgroup $H$ the resulting map $X^H\to Y^G$ is a weak equivalence (resp. fibration). As generating cofibrations you can use the inclusions $G/H\times S^{n-1}\subset G/H\times D^n$ for all $H$ and $n$. In particular $G$-CW complexes are cofibrant. And strong homotopy equivalence in the model category sense becomes equivariant homotopy equivalence in the obvious sense. Thus the statement above becomes an instance of the general principle that for cofibrant objects every weak equivalence is a strong equivalence.

By the way, a reasonable question is, is the analogous statement true with orbits instead of fixed points? That's not true, though. Take a CW space $Z$ that is acyclic but not contractible. Let $X$ be the suspension of $Z$, and let $G$ of order $2$ act on it by switching the two cones, with $Z$ as fixed point set. Then both $X$ and the orbit space are contractible, but $X$ is not equivariantly contractible because the fixed point set is not contractible.