We investigate finite-width optical-beam propagation in a medium with uniform torsion described by the geometric theory of a continuous distribution of screw dislocations. Starting from the Riemann-Cartan framework that yields torsion-induced circular birefringence for local plane waves, we construct a minimal paraxial beam model in which the same contortion-driven helicity splitting remains explicit. The resulting beam equation is used as a controlled effective paraxial reduction, not as an exact finite-beam solution of the complete Riemann-Cartan Maxwell problem. Its validity assumes a predominantly forward beam $k_0{w}_0\gg 1$, weak effective optical detuning $|{\mathrm{\Omega }}_{\mathrm{o}}|w_0\ll k_0$ (equivalently $|{\mathrm{\Omega }}_{\mathrm{g}}|w_0\ll 1$), and slowly varying helicity phases over the transverse scale. We show that uniform torsion breaks the degeneracy between the two circular-polarization sectors and induces a geometry-induced helicity-dependent propagation phase that scales with both the propagation distance and the radial position in the beam. As a consequence, a finite-width beam develops spatially varying polarization textures across its transverse profile, naturally described by the Stokes parameters. We introduce beam-level observables based on the integrated Stokes vector, the transverse inhomogeneity of the polarization texture, and the number of resolved radial polarization domains, thereby connecting the torsion parameter to experimentally accessible beam diagnostics. The paper combines two complementary levels of description: an analytic short-distance regime, used to isolate the geometric mechanism, and full paraxial propagation including diffraction, used to test the robustness of the predicted textures. Within the cylindrically symmetric minimal model, the most robust structured-light signature of uniform torsion is beam-resolved polarization structuring, whereas strong orbital-angular-momentum conversion is not expected without additional azimuthal structure. We also discuss a deliberately extended, $q$-plate-like effective model only to identify the additional azimuthal geometric connection required for spin-orbit conversion; this extension is not presented as an intrinsic prediction of a uniform-torsion medium.

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