Polarimetry is a powerful tool for astrophysical observations that has yet to be exploited in the X-ray band. For satellite-borne and sounding rocket experiments, we have developed a photoelectric gas polarimeter to measure X-ray polarization in the 2-10 keV range utilizing a time projection chamber (TPC) and advanced micro-pattern gas electron multiplier (GEM) techniques. We carried out performance verification of a flight equivalent unit (1/4 model) which was planned to be launched on the NASA Gravity and Extreme Magnetism Small Explorer (GEMS) satellite. The test was performed at Brookhaven National Laboratory, National Synchrotron Light Source (NSLS) facility in April 2013. The polarimeter was irradiated with linearly-polarized monochromatic X-rays between 2.3 and 10.0 keV and scanned with a collimated beam at 5 different detector positions. After a systematic investigation of the detector response, a modulation factor ≥35% above 4 keV was obtained with the expected polarization angle. At energies below 4 keV where the photoelectron track becomes short, diffusion in the region between the GEM and readout strips leaves an asymmetric photoelectron image. A correction method retrieves an expected modulation angle, and the expected modulation factor, ∼20% at 2.7 keV. Folding the measured values of modulation through an instrument model gives sensitivity, parameterized by minimum detectable polarization (MDP), nearly identical to that assumed at the preliminary design review (PDR).