Three-dimensional (3D) bioprinting is an additive manufacturing process in which the combination of biomaterials and living cells, referred to as a bioink, is deposited layer-by-layer to form biologically active 3D tissue constructs. Recent advancements in the field show that the success of this technology requires the development of novel biomaterials or the improvement of existing bioinks. Polyethylene glycol (PEG) is one of the well-known synthetic biomaterials and has been commonly used as a photocrosslinkable bioink for bioprinting; however, other types of cell-friendly crosslinking mechanisms to form PEG hydrogels need to be explored for bioprinting and tissue engineering. In this work, we proposed micro-capillary based bioprinting of a novel molecularly engineered PEG-based bioink that transiently incorporates low molecular weight gelatin (LMWG) fragments. The rheological properties and release profile of the LMWG fragments were characterized, and their presence during hydrogel formation had no effect on the swelling ratio or sol fraction when compared to PEG hydrogels formed without the LMWG fragments. For bioprinting, PEG was first functionalized with cell-adhesive RGD ligands and was then crosslinked using protease-sensitive peptides via a Michael-type addition reaction inside the micro-capillary. The printability was assessed by the analysis of extrudability, shape fidelity, and printing accuracy of the hydrogel filaments after the optimization of the gelation conditions of the PEG-based bioink. The LMWG fragments supplemented into the bioink allowed the extrusion of smooth and uniform cylindrical strands of the hydrogel and improved shape fidelity and printing accuracy. Encapsulated cells in both bioprinted and non-bioprinted PEG-based hydrogels showed high viability and continued to proliferate over time in culture with a well-defined cell morphology depending on the presence of the cell adhesive peptide RGD. The presented micro-capillary based bioprinting process for a novel PEG-based bioink can be promising to construct complex 3D structures with micro-scale range and spatiotemporal variations without using any cytotoxic photoinitiator, UV light, or polymer support.