Three-dimensional (3D) bioprinting of hydrogel-based constructs at adequate consistency and reproducibility can be obtained through a compromise between the hydrogel's inherent instability and printing fidelity. There is an increasing demand to develop bioprinting modalities that enable high-fidelity fabrication of 3D hydrogel structures that closely correspond to the envisioned design. In this work, we performed a systematic, in-depth characterization and optimization of embedded 3D bioprinting to create 3D gelatin-methacryloyl (gelMA) structures with highly controlled fidelity using Carbopol as suspension bath. The role of various embedded printing process parameters in bioprinting fidelity was investigated using a combination of experimental and theoretical approaches. We examined the effect of rheological properties of gelMA and Carbopol at varying concentrations, as well as printing conditions on the volumetric flow rate of gelMA bioink. Printing speed was examined and optimized to successfully print gelMA into the support bath at varying Carbopol concentrations. Printing fidelity was characterized in terms of printed strand diameter, uniformity, angle, and area. The optimal Carbopol solution that retained filament shape at highest fidelity was determined. The efficacy of developed bioprinting approach was then demonstrated by fabricating 3D hydrogel constructs with varying geometries and visualized using an advanced synchrotron-based imaging technique. We also investigated the influence of the Carbopol medium on cross-linking and the resulting stiffness of gelMA constructs. Finally, in vitro cytotoxicity of the developed bioprinting approach was assessed by printing human umbilical vein endothelial cells encapsulated in the gelMA bioink. These results demonstrate the significance of the close interplay between bioinksupport bath rheology and printing parameters and help to establish an optimized workflow for creating 3D hydrogel structures with high fidelity and cytocompatibility via embedded bioprinting techniques. This robust platform could further expand the application of bioprinted soft tissue constructs in a wide variety of biomedical applications.