Vector optimization problems are a generalization of multiobjective optimization in which the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative octant. Due to its real life applications, it is important to have practical solution approaches for computing. In this work, we consider the inexact projected gradient-like method for solving smooth constrained vector optimization problems. Basically, we prove global convergence of any sequence produced by the method to a stationary point assuming that the objective function of the problem is $K$-quasiconvex, instead of the stronger $K$-convexity assumed in the literature.