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Cell shape changes, driven by the contractile actomyosin cortex, are essential for multicellular organisms. Yet, how contractility and cell shape changes emerge from molecular scale interactions between proteins in the actomyosin cortex remains unclear. Oocytes of the bat star Patiria miniata exhibit a traveling wave of cellular deformation during meiosis, known as a surface contraction wave (SCW). Here, we exploit this highly stereotypical deformation to study how cortical properties are set by microscale processes. By pharmacologically modulating the levels of polymerized actin, we show that the wildtype composition maximizes the cellular deformation rate. To understand these results, we developed an active fluid model for shape changes that demonstrates that deformation rates are set by the ratio between cortical viscosity and active contractile stresses and provides a framework for deriving these two key material properties from coarse-grained molecular-scale interactions. In this model, the ratio of active stress to viscosity peaks at intermediate admixtures of passive crosslinkers. This explains the observations of the drug treatment experiments and makes additional predictions, namely that overexpression of either myosin or passive crosslinkers would decrease deformation rates, which we verified experimentally. Together, this shows that the interplay between cortical active stress and viscosity relies on balancing the ratio of myosin to passive crosslinkers on each filament, which can be modulated by actin density. Our results argue that changing the relative molecular compositions of cortical crosslinks and motors provides a mechanism that biological systems could exploit for robust and predictable control over both viscosity and active stresses.