Galactic short period double white dwarfs (DWD) are guaranteed gravitational wave (GW) sources for the next generation of space-based interferometers sensitive to low-frequency GWs (10-4- 1 Hz). Here we investigate the possibility of constraining the white dwarf (WD) properties through measurements of apsidal precession in eccentric binaries. We analyze the general relativistic (GR), tidal, and rotational contributions to apsidal precession by using detailed He WD models. We find that apsidal precession can lead to a detectable shift in the emitted GW signal, the effect being stronger (weaker) for binaries hosting hot (cool) WDs. We find that in hot (cool) DWDs tides dominate the precession at orbital frequencies above ~0.01 mHz (~1 mHz). Analyzing the apsidal precession of these sources only accounting for GR would potentially lead to an extreme overestimate of the component masses. Finally, we derive a relation that ties the radius and apsidal precession constant of cool WD components to their masses, therefore allowing tides to be used as an additional mass measurement tool.