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By one-dimensional particle-in-cell (PIC) simulations, the propagation and stability of relativistic electromagnetic (EM) solitary waves as well as modulational instability of plane EM waves are studied in uniform cold electron-ion plasmas. The investigation not only confirms the solitary wave motion characteristics and modulational instability theory, but more importantly, gives the following findings. For a simulation with the plasma density 1023 m?3 and the dimensionless vector potential amplitude 0.18, it is found that the EM solitary wave can stably propagate when the carrier wave frequency is smaller than 3.83 times of the plasma frequency. While for the carrier wave frequency larger than that, it can excite a very weak Langmuir oscillation, which is an order of magnitude smaller than the transverse electron momentum and may in turn modulate the EM solitary wave and cause the modulational instability, so that the solitary wave begins to deform after a long enough distance propagation. The stable propagation distance before an obvious observation of instability increases (decreases) with the increase of the carrier wave frequency (vector potential amplitude). The study on the plane EM wave shows that a modulational instability may occur and its wavenumber is approximately equal to the modulational wavenumber by Langmuir oscillation and is independent of the carrier wave frequency and the vector potential amplitude. This reveals the role of the Langmuir oscillation excitation in the inducement of modulational instability and also proves the modulational instability of EM solitary wave.