If you accept the Whitehead theorem, then simple obstruction theory (without cohomology) determines the homotopy type of spaces with homotopy groups in only one dimension.  That is, 
you can easily show that if the only nonzero homotopy group of the CW complex $Y$ 
is in dimension $n$, then for any $(n-1)$-connected CW complex
$$
[X, Y] \cong \mathrm{Hom}( \pi_n(X), \pi_n(Y) ) .
$$
Then if the homotopy groups of $X$ are also concentrated in dimension $n$, 
an isomorphism $\pi_n(X) \xrightarrow{\cong} \pi_n(Y)$ gives rise to 
a homotopy equivalence $X\xrightarrow{\simeq} Y$.

But what if we don't want to use Whitehead?  I think you are stuck, because basic 
(model category theoretic) arguments can be used to prove Whitehead (at least for 
CW complexes with finitely many finitely generated homotopy groups?) from the uniqueness
of Eilenberg-Mac Lane spaces. 

(Prove it by induction for CW complexes with at most $n$ nonzero homotopy groups;  this requires us to convert maps to fibrations, and we don't want to end up with non-CW complexes, which is why I said we might need finitely generated homotopy groups.)
 
CLARIFICATION:  When I said "model-theoretic" I meant the basic facts about cofibrations, 
fibrations, induced maps, etc.  The finite generation comes in when we convert a map to a fibration:  this is done with a path space and a pullback, and these things can easily take us away from CW complexes, but my plan was to impose conditions that would guarantee we can apply Milnor's theorem (on loops of CW complexes and so on) to the CW complexes in question.