We present the cosmological analysis of 752 photometrically classified Type Ia Supernovae (SNe Ia) obtained from the full Sloan Digital Sky Survey II (SDSS-II) Supernova (SN) Survey, supplemented with host-galaxy spectroscopy from the SDSS-III Baryon Oscillation Spectroscopic Survey. Our photometric-classification method is based on the SN classification technique of Sako et al. (2011, Cat. J/ApJ/738/162), aided by host-galaxy redshifts (0.05<z<0.55). SuperNova ANAlysis simulations of our methodology estimate that we have an SN Ia classification efficiency of 70.8%, with only 3.9% contamination from core-collapse (non-Ia) SNe. We demonstrate that this level of contamination has no effect on our cosmological constraints. We quantify and correct for our selection effects (e.g., Malmquist bias) using simulations. When fitting to a flat {Lambda}CDM cosmological model, we find that our photometric sample alone gives {Omega}_m_=0.24^+0.07^_-0.05_ (statistical errors only). If we relax the constraint on flatness, then our sample provides competitive joint statistical constraints on {Omega}_m_ and {Omega}_{Lambda}_, comparable to those derived from the spectroscopically confirmed Three-year Supernova Legacy Survey (SNLS3). Using only our data, the statistics-only result favors an accelerating universe at 99.96% confidence. Assuming a constant wCDM cosmological model, and combining with H_0_, cosmic microwave background, and luminous red galaxy data, we obtain w=-0.96^+0.10^_-0.10_, {Omega}_m_=0.29^+0.02^_-0.02_, and {Omega}_k_=0.00^+0.03^_-0.02_ (statistical errors only), which is competitive with similar spectroscopically confirmed SNe Ia analyses. Overall this comparison is reassuring, considering the lower redshift leverage of the SDSS-II SN sample (z<0.55) and the lack of spectroscopic confirmation used herein. These results demonstrate the potential of photometrically classified SN Ia samples in improving cosmological constraints.