<p>We introduce a new analytical approach to computing the gravitational potential of a homogeneous tetrahedron, a fundamental building block in polyhedral modeling. This formalism is entirely free of numerical singularities. Unlike many existing solutions for polyhedral bodies, which are valid only outside the body, our expressions are uniformly applicable across the interior, boundary, and exterior of the tetrahedron. The method eliminates singularities by explicitly resolving geometric and analytical irregularities near edges, vertices, and face planes. We provide implementations in Python, MATLAB, and Julia, together with algorithmic pseudocode to facilitate adoption. Numerical validation via continuity tests and a Laplacian test confirms the stability, precision, and consistency of the formulation across all spatial regimes. This work broadens the class of analytically tractable gravitational primitives, complementing our prior solution for the cuboid. Since any constant-density polyhedron can be decomposed into tetrahedra, the presented formulation establishes a general pathway for modeling the gravitational fields of bodies with arbitrary geometry, providing robust tools for geodetic and geophysical modeling.</p>