The hierarchical formation scenario for the stellar halo requires the accretion and disruption of dwarf galaxies, yet low-metallicity halo stars are enriched in a-elements compared to similar, low-metallicity stars in dwarf spheroidal (dSph) galaxies. We address this primary challenge for the hierarchical formation scenario for the stellar halo by combining chemical evolution modeling with cosmologically motivated mass accretion histories for the Milky Way dark halo and its satellites. We demonstrate that stellar halo and dwarf galaxy abundance patterns can be explained naturally within the CDM framework. Our solution relies fundamentally on the CDM model prediction that the majority of the stars in the stellar halo were formed within a few relatively massive, 65 x 10 super(10) M sub( ), dwarf irregular (dIrr) sized dark matter halos, which were accreted and destroyed 610 Gyr in the past. These systems necessarily have short-lived, rapid star formation histories, are enriched primarily by Type II supernovae, and host stars with enhanced a/Fe abundances. In contrast, dwarf dSph galaxies exist within low-mass dark matter hosts of 610 super(9) M sub( ), where supernovae winds are important in setting the intermediate a/Fe ratios observed. Our model includes enrichment from Type Ia and Type II supernovae, as well as stellar winds, and includes a physically motivated supernovae feedback prescription calibrated to reproduce the local dwarf galaxy stellar mass-metallicity relation. We use representative examples of the type of dark matter halos that we expect to host a destroyed "stellar halo progenitor" dwarf, a surviving dIrr, and a surviving dSph galaxy, and show that their derived abundance patterns, stellar masses, and gas masses are consistent with those observed for each type of system. Our model also self-consistently reproduces the observed stellar mass-v sub(circ) relation for local group satellites and produces the correct cumulative mass for the Milky Way stellar halo. We predict that the lowest metallicity stars in intermediate-mass dIrr galaxies such as the SMC and LMC should follow abundance patterns similar to that observed in the stellar halo. Searches for accreted, disrupted, low-mass dwarfs may be enhanced by searching for unbound stars with dSph-like chemical abundance patterns.