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Shapiro time delay is one of the fundamental tests of general relativity and post-Newtonian theories of gravity. Consequently, its measurements can be used to probe the parameter $γ$ which is related to spacetime curvature produced by a unit mass in the post-Newtonian formalism of gravity. To date all measurements of time delay have been conducted on astronomical scales. It was asserted in 2010 that gravitational wave detectors on Earth could be used to measure Shapiro delay on a terrestrial scale via massive rotating systems. Building on that work, we consider how measurements of Shapiro delay can be made using next-generation gravitational wave detectors. We perform an analysis for measuring Shapiro delay with the next-generation gravitational wave detectors Cosmic Explorer and Einstein Telescope to determine how precisely the effect can be measured. Using a rotating mass unit design, we find that Cosmic Explorer and Einstein Telescope can measure the Shapiro delay signal with amplitude signal to noise ratios upwards of $\sim28 $ and $\sim43$ in 1 year of integration time, respectively. By measuring Shapiro delay with this technique, next-generation interferometers will allow for terrestrial measurements of $γ$ in the paramaterized post-Newtonian formalism of gravity with sub-percent precision.