We present the spectrum of baryons in SU(4) gauge theory with fundamental fermion constituents, which is of significant interest for composite dark matter model building. We first compare the spectra and properties of baryons in SU(3) and SU(4) gauge theories (in which they are fermionic and bosonic, respectively) and then compute the cross section for direct detection of dark matter 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 mass of the constituent fermion 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 163×32, 323×64, 483×96, and 643×128 lattices with three different lattice spacings, using Wilson fermions with moderate to large pseudoscalar meson masses. Our results lay a foundation for future analytic and numerical studies of composite baryonic dark matter.