Three-dimensional (3D) bioprinting is an emerging fabrication technique to create 3D constructs with living cells. Notably, bioprinting bioinks are limited due to the mechanical weakness of natural biomaterials and the low bioactivity of synthetic peers. This paper presents the development of a natural bioink from chicken eggwhite and sodium alginate for bioprinting cell-laden patches to be used in endothelialized tissue engineering applications. Eggwhite was utilized for enhanced biological properties, while sodium alginate was used to improve bioink printability. The rheological properties of bioinks with varying amounts of sodium alginate were examined with the results illustrating that 2.0–3.0% (w/v) sodium alginate was suitable for printing patch constructs. The printed patches were then characterized mechanically and biologically, and the results showed that the printed patches exhibited elastic moduli close to that of natural heart tissue (20–27 kPa) and more than 94% of the vascular endothelial cells survived in the examination period of one week post 3D bioprinting. Our research also illustrated the printed patches appropriate water uptake ability (>1800%).