This work is devoted to the study of the asymptotics of the gravitational field of primary particles with nonzero rest mass. These particles are structurally composed of two closed "null strings" (thin closed tubes of a massless scalar field) in the shape of a circle, and they are formed in a gas of null strings as a result of gravitational interaction. It is shown that on time scales much larger than the time of one complete cycle of oscillation of the null strings forming a particle, or at distances much larger than the dimensions of the region within which the oscillations of interacting null strings occur, the gravitational field of such a particle is described by the Minkowski metric. It is noted that with decreasing observation time or on distance scales that are commensurate with the size of primary particles, significant deviations of the gravitational field from the flat Minkowski space-time should appear in the gas of null strings.