Proton recoil polarization was measured in the quasielastic 4He(e,ep)3H reaction at Q^2 = 0.8 (GeV/c)^2 and 1.3 (GeV/c)^2 with unprecedented precision. The polarization-transfer coefficients are found to differ from those of the 1H(e,e p) reaction, contradicting a relativistic distorted-wave approximation, and favoring either the inclusion of medium-modified proton form factors predicted by the quark-meson coupling model or a spin-dependent charge-exchange final-state interaction. For the first time, the polarization-transfer ratio is studied as a function of the virtuality of the proton.