At present information about viscous properties of the eye's sclera is scarce, the main reason being that direct measurements cause technical problems. Nevertheless, coefficients of the sclera viscosity are of paramount interest for biomechanics and ophthalmologists. This paper investigates a method for determining the shear viscosity of the sclera based on a comparison of results from mathematical modeling with estimated experimental data from measurements of the IntraOcular Pressure (IOP) over a long time after an intravitreal injection (injection into the vitreous body of the eyeball) into enucleated eyes. In order to model the IOP behavior we consider a viscoelastic spherical layer subjected to a centrally symmetric load as follows. The excess pressure is considered, hence, the external pressure is absent and a displacement of the inner boundary is specified in order to take the intravitreal injection volume into account. We assume that during the experiment the volume of the eyeball remains constant and that the material of the sclera obeys linear transversely isotropic elasticity. The function for the IOP, which is based on fitted experimental data presented in the literature, decreases up to a limit value. As a result, we obtain a value for the shear viscosity of the sclera corresponding to the best adjustment between experimental data and theoretical prediction.