Kinetic theory has shown that in normal scale-invariant systems, the shear viscosity is proportional to the pressure and their ratio is the relaxation time. This is an example of a relation connecting thermodynamic quantities (the pressure) and transport coefficients (the shear viscosity). The unitary Fermi gas, consisting of two-component fermions on the verge of forming two-body bound states, is an example of scale-invariant systems because of its divergent two-body scattering length. At low temperatures, the presence of superfluidity complicates the relation. By implementing a gauge-invariant linear response theory, we found the shear viscosity is related not only to the pressure and relaxation time, but also to the superfluid density and an additional response function from the shear momentum transfer via the Cooper pairs. We have tested the relation with and without pairing fluctuations which are crucial in describing the BCS-BEC crossover as the attractive interaction increases. The relation is exact for the BCS superfluid, and we propose an approximate relation for superfluids with pairing fluctuations. Moreover, the relation works in the presence of population imbalance between the two components.