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We use a version of the instant-form relativistic quantum mechanics of composite systems to obtain the gravitational form factors of the pion in a common approach to its electroweak and gravitational properties. In the preceding work [arXiv:2010.11640] we formulated the mathematical background, presented the principal scheme of calculation and testified the obtained qualitative results to satisfy the general constraints given by the principles of the theory of hadron structure. In the present work we give the detailed calculation of the gravitational form factors in large range of momentum transfer, their static limits and the slopes at zero value, the mean-square mass and mechanical radii of the pion. Now we take into account the qravitational structure of the constituent quarks. We show that the results are almost insensitive to the type of the model two-quark wave function in a close analogy to the case of the pion electromagnetic form factor. We present a correct calculation of the form factor $D$ and corresponding matrix element of the energy-momentum tensor, going beyond the scope of the modified impulse approximation. Most of the parameters that we use for the calculation had been fixed even earlier in our works on the pion electromagnetic form factors. The only free parameter is the $D$-term of the constituent quark, which we fix by fitting the result for the slope at zero of the normalized to pion $D$-term form factor $D$ of pion, to a choosen experimental value.