The design method to obtain broadband, single-layer, microstrip antennas with circularly polarized (CP) tilted-beam radiation at a customized angle under a single-feed point is presented. The microstrip antenna topology is a refractive index-modulated parasitic array where a coaxial probe feeds the first element and others are excited through surface-wave coupling. In the design process, first, a novel type of folded split ring resonator (FSRR) is specially engineered to serve as the basic elements that could provide two neighboring orthogonal modes within a miniaturized size. Second, to ensure each mode of the FSRRs radiates its correspondent far-field component, e.g., E_θ or E_φ with the same amplitude, which is critical to CP radiation, the coupling coefficients of two coupled FSRRs are extracted to quantitatively determine the required edge-to-edge spacing of two neighboring FSRRs. Third, to achieve tilted-beam radiation at a predefined angle, the index-modulation principle is applied to calculate the specific dimensions of each index-modulated FSRR. Ultimately, the developed CP tilted-beam antenna exhibits a broad bandwidth of 11.57% at the predefined tilt angle of 30°. Advantages including broad bandwidth, single feed, compactness, and single-layer configuration, indicate its great potential for applications such as satellite communications and indoor base stations.