The virial stress is the most commonly used definition of stress in discrete molecular systems. This stress includes two parts. The first part depends on the mass and velocity of atomic particles, reflecting an assertion that mass transfer causes mechanical stress to be applied on stationary spatial surfaces external to an atomic particle system. The second part depends on interatomic forces and atomic positions, providing a continuum measure for the internal mechanical interactions between particles. For the simple conditions of rigid translation and uniform tension, the virial stress yields apparently erroneous interpretations. It is shown that the virial stress violates balance of momentum and does not possess physical significance as a measure for mechanical interaction between material points. The lack of physical significance is both at the individual atom level in a time-resolved sense and at the system level in a statistical sense. As a stress-like quantity, the virial stress has the geometric interpretation of being a measure for momentum change in a fixed spatial region. It is demonstrated that the interatomic force term alone is a valid stress measure and can be identified with the Cauchy stress. The discussions here focus on an error in the theoretical derivation of the virial stress that led to the inclusion of the kinetic energy term and on the conceptual flaws in the argument commonly used for including it. The result here demonstrates that, for gas systems in macroscopic steady state, the isotropic statistical average of the internal stress tensor over time and space is not equal to the negative of the macroscopic pressure tensor for the system. The statistical average of the stress is equal to the negative of the pressure only at absolute zero when all molecular motions cease.