  If the weak equivalence principle (WEP) is broken, the measured values of the
parametrized post-Newtonian parameter $\gamma$ from photons with left- and
right-handed circular polarizations should differ slightly, leading to the
arrival-time difference of these two circular components. Thus, the
polarization vector of a linearly polarized light may rotate during the
propagation. The rotation angle of the polarization vector depends on both the
photon energy and the distance of the source. It is believed that if the
rotation angle differs by more than $\pi/2$ over an energy range, then the net
polarization of the signal would be significantly suppressed and could not be
as high as the observed level. Thus, the detection of highly polarized photons
implies that the relative rotation angle ($\Delta\Theta$) should not be too
large. In this paper, we give a detailed calculation on the evolution of
gamma-ray burst (GRB) polarization arising from a possible violation of the
WEP, and we find that more than $60\%$ of the initial polarization degree can
be conserved even if $\Delta\Theta$ is larger than $\pi/2$. In addition, to
tightly constrain the WEP violation, GRBs with harder spectra and polarization
observations in a wider energy range seem to be favored. Applying our formulas
to the measurements of linear polarization from GRB 110721A and GRB 061122, we
obtain strict limits on the differences of the $\gamma$ values as low as
$\Delta\gamma<1.3\times10^{-33}$ and $\Delta\gamma<0.8\times10^{-33}$. These
provide the most stringent limits to date on a deviation from the WEP,
improving at least 6 orders of magnitude over previous bounds.
