Sagittarius A* (Sgr A*) is the variable radio, near-infrared (NIR), and X-ray source associated with accretion onto the Galactic center black hole. We present an analysis of the most comprehensive NIR variability data set of Sgr A* to date: eight 24 hr epochs of continuous monitoring of Sgr A* at 4.5{mu}m with the IRAC instrument on the Spitzer Space Telescope, 93 epochs of 2.18{mu}m data from Naos Conica at the Very Large Telescope, and 30 epochs of 2.12{mu}m data from the NIRC2 camera at the Keck Observatory, in total 94929 measurements. A new approximate Bayesian computation method for fitting the first-order structure function extracts information beyond current fast Fourier transformation (FFT) methods of power spectral density (PSD) estimation. With a combined fit of the data of all three observatories, the characteristic coherence timescale of Sgr A* is {tau}_b_=243_-57_^+82^min (90% credible interval). The PSD has no detectable features on timescales down to 8.5min (95% credible level), which is the ISCO orbital frequency for a dimensionless spin parameter a=0.92. One light curve measured simultaneously at 2.12 and 4.5{mu}m during a low flux-density phase gave a spectral index {alpha}_s_=1.6+/-0.1 (F_{nu}_{propto}{nu}^-{alpha}_s_^). This value implies that the Sgr A* NIR color becomes bluer during higher flux-density phases. The probability densities of flux densities of the combined data sets are best fit by log-normal distributions. Based on these distributions, the Sgr A* spectral energy distribution is consistent with synchrotron radiation from a non-thermal electron population from below 20GHz through the NIR.