The J/ψ pT spectrum and nuclear modification factor (RAA) are reported for pT<5GeV/c and |y|<1 from 0% to 60% central Au+Au and Cu+Cu collisions at sNN=200GeV at STAR. A significant suppression of pT-integrated J/ψ production is observed in central Au+Au events. The Cu+Cu data are consistent with no suppression, although the precision is limited by the available statistics. RAA in Au+Au collisions exhibits a strong suppression at low transverse momentum and gradually increases with pT. The data are compared to high-pT STAR results and previously published BNL Relativistic Heavy Ion Collider results. Comparing with model calculations, it is found that the invariant yields at low pT are significantly above hydrodynamic flow predictions but are consistent with models that include color screening and regeneration.