= Solution
Assume the test body moves slowly, $dx^i/dt=O(\epsilon)$, and that the field is quasistatic, so time derivatives are of higher order. The spatial <geodesic equation>, parametrized by coordinate time, then reduces at leading order to
$$
\frac{d^2x^i}{dt^2}=-\Gamma^i{}_{00}
=\frac12\partial_i h_{00}+O(\epsilon^3).
$$
Defining the Newtonian potential by
$$
\boxed{h_{00}=-2\Phi}
$$
gives
$$
\boxed{\frac{d^2\mathbf x}{dt^2}=-\boldsymbol\nabla\Phi},
$$
the Newtonian equation of motion. Terms involving $h_{0i}$, spatial velocity, or time derivatives enter beyond the stated order.
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