We report the growth of InAs/GaSb core-shell heterostructure nanowires with smooth sidewalls on Si substrates using metal-organic chemical vapor deposition with no assistance from foreign catalysts. Sb adatoms were observed to strongly influence the morphology of the GaSb shell. In particular, Ga droplets form on the nanowire tips when a relatively low TMSb flow rate is used, whereas the droplets are missing and the radial growth of the GaSb is enhanced due to a reduction in the diffusion length of the Ga adatoms when the TMSb flow rate is increased. Moreover, transmission electron microscopy measurements revealed that the GaSb shell coherently grew on the InAs core. The results obtained here show that the InAs/GaSb core-shell nanowires grown using the Si platform have strong potential in the fabrication of future nanometer-scale devices and in the study of fundamental quantum physics.