Abstract: We present a study of the stellar populations of globular clusters (GCs) in the Virgo Cluster core with a homogeneous spectroscopic catalog of 692 GCs within a major-axis distance R (maj) = 840 kpc from M87. We investigate radial and azimuthal variations in the mean age, total metallicity, [Fe/H], and alpha-element abundance of blue (metal-poor) and red (metal-rich) GCs using their co-added spectra. We find that the blue GCs have a steep radial gradient in [Z/H] within R (maj) = 165 kpc, with roughly equal contributions from [Fe/H] and [alpha/Fe], and flat gradients beyond. By contrast, the red GCs show a much shallower gradient in [Z/H], which is entirely driven by [Fe/H]. We use GC-tagged Illustris simulations to demonstrate an accretion scenario where more massive satellites (with more metal- and alpha-rich GCs) sink further into the central galaxy than less massive ones, and where the gradient flattening occurs because of the low GC occupation fraction of low-mass dwarfs disrupted at larger distances. The dense environment around M87 may also cause the steep [alpha/Fe] gradient of the blue GCs, mirroring what is seen in the dwarf galaxy population. The progenitors of red GCs have a narrower mass range than those of blue GCs, which makes their gradients shallower. We also explore spatial inhomogeneity in GC abundances, finding that the red GCs to the northwest of M87 are slightly more metal-rich. Future observations of GC stellar population gradients will be useful diagnostics of halo merger histories.