Calculating from a given action the energy-momentum tensor $ \tilde{T}_{\mu \nu} $ (differentiating respect to $ \delta g^{\mu \nu}) $ I can create gravity by a generalization of the Einstein field equation? $$ G_{\mu \nu}=\frac{8 \pi G}{c^4} \tilde{T}_{\mu \nu}\tag{1} $$ where $\tilde{T}_{\mu \nu}$ is different from the matter energy-momentum tensor.
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Really you should be thinking about the total action, $$S = S_{EH}[g] + S_{M}[g,\phi^A] \ ,$$ which includes both the matter action $S_{M}[g,\phi^A]$ and the Einstein-Hilbert Action $S_{EH}[g]$. Then the variation of the total action with respect to the metric $g_{\mu \nu}$ gives the gravitational field equations, $G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}$, (see the wiki link for the details). But yes, this leads to the equations you wrote down.
Small note: we vary the matter action/Lagrangian with respect to the metric, not differentiate.
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