(1) Background: Concentric ring artifacts in reconstructed SPECT images indicate the presence of detector non-uniformity in gamma camera systems. The identification of these artifacts is generally visual and not quantitative. The aim of our study was to evaluate observer assessments of the presence of concentric rings in reconstructed SPECT phantom images and to verify whether quantitative texture analysis can detect such artifacts, which are detrimental to accurate tumor detection. (2) Methods: Test data were acquired as part of the quarterly quality assurance program using a standardized SPECT phantom containing solid spheres, solid rods, and a water solution of 99mTc. Forty separate SPECT acquisitions were analyzed to assess the presence of ring artifacts. Two experienced medical physicists independently reviewed transaxial images and graded the severity of artifacts on a five-point scale. Quantitative radiomic features were computed for volumes of interest located in the uniform phantom section. In addition to these, radial contrast (RContrast) and radial root-mean-square contrast (RRMSC) were also calculated and derived from the radial profile of summed slices transformed into polar coordinates. (3) Results: Artifacts were considered sufficiently severe to warrant camera re-tuning in 10 rod sections, 17 sphere sections, and 16 uniform sections. In the uniform sections, there was “good agreement” for inter-observer and intra-rater assessments (κ = 0.66, Fisher exact p < 0.0001 and κ = 0.61, and Fisher exact p = 0.001, respectively). The two radial features agreed significantly (p < 0.001) with visual severity judgment of ring artifacts in uniform sections and were selected as informative about the presence of ring artifacts by LASSO approach. The increased magnitude of RContrast and RRMSC correlated significantly with increasingly severe artifact scores (ρ = 0.65–0.66, p < 0.0001). (4) Conclusions: There was good agreement between the physicists with respect to the presence of circular ring artifacts in uniform sections of SPECT quality assurance scans, with the artifacts accurately detected by radial contrast and noise-to-signal ratio measurements.