The spectra of several high-redshift (z>6) quasars have shown evidence for a Gunn-Peterson (GP) damping wing, indicating a substantial mean neutral hydrogen fraction (x_HI > 0.03) in the z ~ 6 intergalactic medium (IGM). However, previous analyses assumed that the IGM was uniformly ionized outside of the quasars HII region. Here we relax this assumption and model patchy reionization scenarios for a range of IGM and quasar parameters. We quantify the impact of these differences on the inferred x_HI, by fitting the spectra of three quasars: SDSS J1148+5251 (z=6.419), J1030+0524 (z=6.308), and J1623+3112 (z=6.247). We find that the best-fit values of x_HI in the patchy models agree well with the uniform case. More importantly, we confirm that the observed spectra favor the presence of a GP damping wing, with peak likelihoods decreasing by factors of > few - 10 when the spectra are modeled without a damping wing. We also find that the Ly alpha absorption spectra, by themselves, cannot distinguish the damping wing in a relatively neutral IGM from a damping wing in a highly ionized IGM, caused either by an isolated neutral patch, or by a damped Ly alpha absorber (DLA). However, neutral patches in a highly ionized universe (x_HI < 0.01), and DLAs with the large required column densities (N_HI > few x 10^{20} cm^{-2}) are both rare. As a result, when we include reasonable prior probabilities for the line of sight (LOS) to intercept either a neutral patch or a DLA at the required distance of ~ 40-60 comoving Mpc away from the quasar, we find strong lower limits on the neutral fraction in the IGM, x_HI > 0.1 (at 95% confidence). This strengthens earlier claims that a substantial global fraction of hydrogen in the z~6 IGM is in neutral form.
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