In this work, we investigate whether experimental surface tension measurements, which are less sensitive to quantum and self-polarization corrections, are able to replace the usual reliance on the heat of vaporization as experimental reference data for fitting force field models of molecular liquids. To test this hypothesis, we develop the fitting protocol necessary to utilize surface tension measurements in the ForceBalance optimization procedure to determine revised parameters for both three-point and four-point water models TIP3P-ST and TIP4P-ST. We find that the incorporation of surface tension in the fit results in a rigid three-point model that reproduces the correct temperature of maximum density of water for the first time but also leads to overstructuring of the liquid and less accurate transport properties. The rigid four-point TIP4P-ST model is highly accurate for a broad range of thermodynamic and kinetic properties, with similar performance compared to recently developed four-point water models. The results show surface tension to be a useful fitting property in general, especially when self-polarization corrections or nuclear quantum corrections are not readily available for correcting the heat of vaporization as is the case for other molecular liquids.