I think the answer is no, due to [this answer][1] to [this question][2]. There are Moore spaces with the same cohomology rings and module structure over the mod $p$ Steenrod algebra, which nevertheless have different homology groups and so their singular complexes cannot be weakly equivalent.


  [1]: http://mathoverflow.net/questions/55365/counterexamples-in-algebraic-topology/55375#55375
  [2]: http://mathoverflow.net/questions/55365/counterexamples-in-algebraic-topology