We present EMIR K band multi-object spectroscopy with the Gran Telescopio Canarias of 2 out of 6 of the X-ray binary (XRB) candidates in the Galactic Center (GC) discovered by Garner et al. (in prep). We report the detection of CO bands in both of these sources, confirming the prediction by Garner et al. (in prep) that they are red giants or red supergiants (RSGs). We also find double peaked velocity shifted Brackett gamma emission in one source, a classic signature of an accretion disk around a compact object. Using the measured velocity of the Brackett gamma emission line and the measured radial velocity of the counterpart, we calculate a minimum mass for the compact object of >=~8 solar masses (with various assumptions, P = 215 days, i = 90 deg, e<=0.5) indicating a black hole as the likely compact object. If confirmed, this is the first dynamical mass measurement of a black hole XRB in the GC. Another source shows an extremely high radial velocity of ~-360 km/s, indicating that it is either a runaway star or a high-amplitude binary around a high-mass compact object. We also present EMIR and FLAMINGOS-2 Ks band infrared spectra of XID 6592, which is potentially the first RSG XRB. We report that XID 6592 no longer shows the Brackett gamma emission line previously seen by DeWitt et al. 2013, and hypothesize that it is potentially a Supergiant Fast X-ray Transient.