Electronic transport through a single-wall metallic carbon nanotube weakly coupled to one ferromagnetic and one nonmagnetic lead is analyzed in the sequential tunneling limit. It is shown that both the spin and charge currents flowing through such systems are highly asymmetric with respect to the bias reversal. As a consequence, nanotubes coupled to one nonmagnetic and one ferromagnetic lead can be effectively used as spin diodes whose functionality can be additionally controlled by a gate voltage.