An excess of gamma rays in the data measured by the Fermi Large Area Telescope in the direction of the Galactic Center has been reported in several publications. This excess, labeled as the Galactic Center excess (GCE), is detected analyzing the data with different interstellar emission models, point source catalogs and analysis techniques. The characteristics of the GCE, recently measured with unprecedented precision, are all compatible with dark matter particles (DM) annihilating in the main halo of our Galaxy, even if other interpretations are still not excluded. We investigate the DM candidates that fit the observed GCE spectrum and spatial morphology. We assume a simple scenario with DM annihilating into a single channel but we inspect also more complicated models with two and three channels. We perform a search for a gamma-ray flux from a list of 48 Milky Way dwarf spheroidal galaxies (dSphs) using state-of-the-art estimation of the DM density in these objects. Since we do not find any significant signal from the dSphs, we put upper limits on the annihilation cross section that result to be compatible with the DM candidate that fits the GCE. However, we find that the GCE DM signal is excluded at the 95% confidence level by the AMS-02 (p) over bar flux data for all purely hadronic (semihadronic) channels unless the diffusive halo size L is smaller than 1.7 kpc (2.6 kpc). Such a small diffusion halo is at the 2 sigma significance lower limit for the results inferred from fluxes of radioactive cosmic rays and is in some tension with results from analyses performed with radio and gamma-ray data. Furthermore, AMS-02 e(+) data rule out the GCE DM interpretation with pure or partial annihilation into e(+)e(-). The only DM candidate that fits the GCE spectrum and fulfills all constraints obtained with the combined dSphs analysis and the AMS-02 (p) over bar and e(+) data annihilates purely (or very dominantly) into mu(+)mu(-), has a mass of similar to 60 GeV and roughly a thermal cross section.