We propose and simulate numerically a permittivity-tunable metamaterial channel, which is composed of alternating layers of graphene and silica. The real part of the permittivity of the proposed metamaterial can be tuned from a positive value to a negative one for a broadband width. Furthermore, optical waves can pass through the metamaterial channel only when its permittivity is tuned to zero. Inspired by this intriguing property of the graphene–silica metamaterial, three basic electro-optical logic gates, including NOT, NOR, and NAND gates, were proposed and numerically investigated by using the finite element method. Taking advantage of the permittivity-tunable property of graphene, the working wavelength of the proposed electro-optical logic gates can be actively controlled by tuning the external voltage applied on the graphene–silica metamaterial. These tunable and ultracompact electro-optical logic gates could benefit the development of nanoscale optical devices for highly integrated photonic circuits.