The geometric fixed point construction can easily be described
in terms of constructions present in any good model for the 
equivariant stable category.  For notational simplicity, I'll
restrict to the case $H = G$.  In general, one can first apply
the change of groups functor from $G$-spectra to $H$-spectra
and then apply the geometric $H$-fixed point functor.  Let
$\mathcal P$ be the family of proper subgroups of $G$. There
is a classifying $G$-space $E\mathcal P$, and we let $\tilde E \mathcal{P}$
denote the cofiber of the evident based $G$-map  $E\mathcal{P}_+ \to S^0$. 
Any good model is tensored over $G$-spaces, so for any $G$-spectrum $X$,
we have the $G$-spectrum $X\wedge \tilde E \mathcal{P}$.  Its categorical
$G$-fixed point spectrum is the geometric fixed point spectrum of $X$.
(This is pointed out briefly in Section XVI.3  of Equivariant Homotopy 
and Cohomology Theory.  I've ignored model theoretic niceties for clarity.)

Edit: In the context of orthogonal $G$-spectra, Mandell and I gave 
a reasonably comprehensive treatment in Section 4 of Chapter V
of  ``Equivariant orthogonal $G$-spectra and $S$-modules'',
all in a more general framework of normal subgroups and
quotient groups.  We give several definitions and comparisons 
(see Lemma 4.15 and Proposition 4.17, including the context-free
one I described above).  The functor preserves cofibrations and 
acyclic cofibrations by Proposition 4.5, but I see no reason to
think it is a left adjoint: I described it in the first place as
a composite of a left and a right adjoint.  The monoidal property is 
Proposition 4.7, and the compatibility with space-level fixed points 
is Corollary 4.6.  The Whitehead theorem is not there, but is 
a comparison with the usual criterion in terms of "categorical"
fixed points (aka Lewis-May fixed points) by the implicit isotropy
separation cofibration and induction.  The term "categorical" refers 
to the fact that this fixed point functor is right adjoint to the functor 
from nonequivariant orthogonal spectra, viewed as $G$-trivial naive 
$G$-spectra, to genuine $G$-spectra obtained by change of universe.  
You first change to the trivial universe and then take level wise fixed points.