We extend the residual-based artificial viscosity (RV), a stabilization technique proven in the finite element setting, and in the high-order summation-by-parts (SBP) setting when applied to scalar equations, to nonlinear systems of conservation laws, focusing on the compressible Euler and in extension the Navier-Stokes equations. We augment the equations with a residual-triggered viscous term that blends a high-order viscosity based on the residual with a first-order upwind cap based on the local maximum wave speed. This leads to a robust viscosity which is only active in the regions where the residual is large, however, remaining asymptotically inactive as the residual vanishes under refinement of the grid. Moreover, boundary conditions are enforced through a mix of projection (strong) and simultaneous-approximation-term (weak) penalties to accommodate slip, inflow/outflow, adiabatic, and no-slip conditions. Accuracy and robustness are demonstrated with third- to ninth-order SBP operators for 2D test-cases. For smooth manufactured solutions of the Euler and Navier-Stokes equations, RV preserves the design order and exhibits errors comparable to the baseline schemes. On the Sod shock tube, RV attains near-optimal first-order L^1 rates and 1/2-order L^2-norm rates. A Riemann problem shows well-captured shocks, contact discontinuities, and expansion fans. For viscous flows, the challenging viscous shock tube of Daru--Tenaud exhibits shock positions and boundary-layer resolution comparable to benchmarks. Where for the artificial viscosity, the high-order residual term dominates except near steep gradients, where the solution is under-resolved. Overall, the residual-based viscosity method RV is shown to combine the stability of first-order upwinding with the accuracy of high-order SBP discretizations, stabilizing shocks and under-resolved features without sacrificing high-order convergence in smooth regions.