New merit functions for variational inequality problems are constructed through the Moreau-Yosida regularization of some gap functions. The proposed merit functions constitute unconstrained optimization problems equivalent to the original variational inequality problem under suitable assumptions. Conditions are studied for those merit functions to be differentiable and for any stationary point of those those functions to be a solution of the original variational inequality problem. Moreover, those functions are shown to provide global error bounds for general variational inequality problems under the strong monotonicity assumption only.