A comprehensive viscous hydrodynamic fit of spectra and elliptic flow for charged hadrons and identified pions and protons from Au+Au collisions of all centralities measured at the Relativistic Heavy Ion Collider is performed and used as the basis for predicting the analogous observables for Pb+Pb collisions at the Large Hadron Collider at sqrt(s)=2.76 and 5.5 A TeV. Comparison with recent measurements of the elliptic flow of charged hadrons by the ALICE experiment shows that the model slightly over-predicts the data if the same (constant) specific shear viscosity eta/s is assumed at both collision energies. In spite of differences in our assumptions for the equation of state, the freeze-out temperature, the chemical composition at freeze-out, and the starting time for the hydrodynamic evolution, our results agree remarkably well with those of Luzum [M. Luzum, Phys. Rev. C 83, 044911 (2011)], indicating robustness of the hydrodynamic model extrapolations. Future measurements of the centrality and transverse momentum dependence of spectra and elliptic flow for identified hadrons predicted here will further test the model and shed light on possible variations of the quark-gluon transport coefficients between RHIC and LHC energies.