  We connect two formalisms that describe deformations away from general
relativity, one valid in the strong-field regime of neutrons stars and another
valid in the radiative regime of gravitational waves: the
post-Tolman-Oppenheimer-Volkoff and the parametrized-post-Einsteinian
formalisms respectively. We find that post-Tolman-Oppenheimer-Volkoff
deformations of the exterior metric of an isolated neutron star induce
deformations in the orbital binding energy of a neutron star binary. Such a
modification to the binding energy then percolates into the gravitational waves
emitted by such a binary, with the leading-order
post-Tolman-Oppenheimer-Volkoff modifications introducing a second
post-Newtonian order correction to the gravitational wave phase. The lack of
support in gravitational wave data for general relativity deformations at this
post-Newtonian order can then be used to place constraints
post-Tolman-Oppenheimer-Volkoff parameters. As an application, we use the
binary neutron star merger event GW170817 to place the constraint $-2.4 \leq
\chi \leq 44$ (at 90% credibility) on a combination of
post-Tolman-Oppenheimer-Volkoff parameters. We also explore the implications of
this result to the possible deformations of the mass-radius relation of neutron
stars allowed within this formalism. This work opens the path towards
theory-independent tests of gravity, combining astronomical observations of
neutron stars and gravitational wave observations.
