Summary: In this paper, we use the dual version of Zorn’s lemma to obtain a minimal fixed point theorem for lower order-preserving set-valued mappings in Hilbert lattices. Applying this fixed point theorem, we introduce an existence theorem of minimal solutions to generalized variational inequalities. Furthermore, we also study the lower order-preservation of solution correspondence for parametric generalized variational inequalities. In contrast to many papers on variational inequalities, our approach is order-theoretic and the results obtained in this paper do not involve any topological continuity with respect to the considered mappings.