A nanoparticle in aqueous solution is modeled as a homogeneous elastic isotropic continuum sphere in contact with an infinite viscous compressible Newtonian fluid. The frequencies and damping of the confined vibrational modes of the sphere are calculated for the material parameters of a CdSe nanoparticle in water and a poly(methyl methacrylate) nanosphere in water. Although the effects of viscosity are found to be negligible for macroscopic objects, for nanoscale objects, both the frequency and damping of the vibrational modes are significantly affected by the viscosity of the liquid. Furthermore, both shear viscosity and bulk viscosity play an important role. A model of the spherical satellite tobacco mosaic virus consisting of outer solid layers with a water core is also investigated, and the viscosity of the water core is found to significantly damp the free vibrational modes. The same approach can be applied for nonspherical geometries and also to viscoelastic nanoparticles.