  Recently, a measurement of the pressure distribution experienced by the
quarks inside the proton has found a strong repulsive (positive) pressure at
distances up to 0.6 femtometers from its center and a (negative) confining
pressure at larger distances. In this paper we show that this measurement puts
significant constraints on modified theories of gravity in which the strength
of the gravitational interaction on microscopic scales is enhanced with respect
to general relativity. We consider the particular case of Eddington-inspired
Born-Infeld gravity, showing that strong limits on $\kappa$, the only
additional parameter of the theory with respect to general relativity, may be
derived from the quark pressure measurement ($|\kappa| \lsim 10^{-1} \, {\rm
m^5 \, kg^{-1} \, s^{-2}}$). Furthermore, we show how these limits may be
significantly improved with precise measurements of the first and second
moments of the pressure distribution inside the proton.
