  A nonlinear Schr\"odinger equation for the envelope of two-dimensional
gravity-capillary waves propagating at the free surface of a vertically sheared
current of constant vorticity is derived. In this paper we extend to
gravity-capillary wave trains the results of \citet{thomas2012pof} and complete
the stability analysis and stability diagram of \citet{Djordjevic1977} in the
presence of vorticity. Vorticity effect on the modulational instability of
weakly nonlinear gravity-capillary wave packets is investigated. It is shown
that the vorticity modifies significantly the modulational instability of
gravity-capillary wave trains, namely the growth rate and instability
bandwidth. It is found that the rate of growth of modulational instability of
short gravity waves influenced by surface tension behaves like pure gravity
waves: (i) in infinite depth, the growth rate is reduced in the presence of
positive vorticity and amplified in the presence of negative vorticity, (ii) in
finite depth, it is reduced when the vorticity is positive and amplified and
finally reduced when the vorticity is negative. The combined effect of
vorticity and surface tension is to increase the rate of growth of modulational
instability of short gravity waves influenced by surface tension, namely when
the vorticity is negative. The rate of growth of modulational instability of
capillary waves is amplified by negative vorticity and attenuated by positive
vorticity. Stability diagrams are plotted and it is shown that they are
significantly modified by the introduction of the vorticity.
