A near-infrared (NIR) survey of 63 radio-selected, ultracompact (UC) H II regions representing 47 different star-forming sites has been completed. The survey was obtained using H-band imaging and moderate-resolution, R = 1200, K-band spectroscopy, centered on the radio emission peak of the UC H II regions. The goal of this survey was to determine the fraction of radio-selected UC H II regions that can be studied with NIR observations and analysis. Approximately 50% of the 63 radio-selected UC H II regions appear to be detected at NIR wavelengths in Brγ emission (107 ergs s-1 cm-2 sr-1). Typical line-of-sight extinction toward the detected UC H II regions ranged from AV = 30 to 50, though one source was measured to have AV = 80. For a few of these UC H II regions, the central ionizing sources are detected through high signal-to-noise ratio NIR spectra of photospheric transitions. This preliminary survey suggests that perhaps 5%-10% of UC H II regions showing NIR counterparts will have directly detectable central ionizing sources. Using the ratio of He I 2.11 to Brγ, the effective temperatures of the central ionizing stars in 25 UC H II regions have been estimated. While He I is not always detected in UC H II regions, when it was found or a meaningful upper limit determined, the spectral type implied by the ratio of He I 2.11 to Brγ closely matched similar estimates of spectral type derived from radio. Model predictions based on mid-infrared measurements appear to underestimate the temperature of the central ionizing stars for which we have directly detected spectral types. The line ratios of H2 2-1 S(1) and 1-0 S(0) relative to the 1-0 S(1) line in our sample of UC H II regions are generally indicative of dense photodissociation regions rather than shocks, similar to what is seen in the Orion Bar. This was true even for UC H II regions showing very weak Brγ emission. While Brγ was generally found to be spatially correlated with the radio emission, H2 showed little correlation with the UC H II regions, typically lying 10'' from the central radio emission. A discussion of each UC H II region studied is included in an extensive appendix.