We present the spectrum of baryons in a new SU(4) gauge theory with fundamental fermion constituents. The spectrum of these bosonic baryons is of significant interest for composite dark matter theories. Here, we compare the spectrum and properties of SU(3) and SU(4) baryons, and then compute the dark-matter direct detection cross section via Higgs boson exchange for TeV-scale composite dark matter arising from a confining SU(4) gauge sector. Comparison with the latest LUX results leads to tight bounds on the fraction of the constituent-fermion mass that may arise from electroweak symmetry breaking. Lattice calculations of the dark matter mass spectrum and the Higgs-dark matter coupling are performed on quenched $16^3 \times 32$, $32^3 \times 64$, $48^3 \times 96$, and $64^3 \times128$ lattices with three different lattice spacings, using Wilson fermions with moderate to heavy pseudoscalar meson masses. Our results lay a foundation for future analytic and numerical study of composite baryonic dark matter.