Fluid injection can promote seismicity by locally elevating pore pressure, thereby reducing effective normal stress and triggering slip on pre-existing faults. In this study, we investigate the role of inertial effects during the nucleation and propagation of fluid-driven shear ruptures on a homogeneous slip-weakening frictional fault, focusing on ruptures that remain ultimately stable. Such ruptures evolve from early-time aseismic growth to large-scale aseismic propagation mediated by the nucleation and arrest of a dynamic event. We assess inertial effects by comparing three-dimensional fully dynamic rupture simulations with quasi-static counterparts [1] using the Spectral Boundary Integral Method. We show that inertial effects do not alter quasi-static stability boundaries but give rise to two key phenomena: dynamic overshoot, whereby rupture extends beyond the quasi-static arrest point, and the radiation of low-amplitude shear waves. We derive semi-analytical estimates of the quasi-static and fully dynamic rupture radii at arrest, whose difference quantifies the dynamic overshoot and is explained by a transfer of kinetic energy into fracture energy. The amplitude of the radiated in-plane shear waves is quantified through scaling analysis validated by simulations. Using a convolution of fault slip, we compute the induced displacements and stresses at remote locations, mimicking measurements in an observatory well. We show that the waves generated by the dynamic rupture are too weak to realistically trigger nearby stressed fault. Overall, our results, obtained for a homogeneous elastic medium under ultimately stable conditions, support the interpretation that microseismicity observed in natural systems is more likely driven by quasi-static stress transfer from aseismic rupture than by dynamic wave-mediated stress transfer. [1] Saez, Alexis, Lecampion, Brice, Fluid-Driven Slow Slip and Earthquake Nucleation on a Slip-Weakening Circular Fault, Journal of the Mechanics and Physics of Solids, 2024