We analyze global entropy solutions of the 2×2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩BVloc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L1 ∩ L∞ ∩ BVloc perturbation of the Riemann initial data, provided that the corresponding solutions are in L∞ and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L∞ with arbitrarily large oscillation.