The biggest point is of course that all motion is relative, as many have pointed out. Even barring that, pretending that we live in Copernican times and assuming that the Sun is the center of the universe, the question posits that the force of gravity might be proportional to mass and velocity. This would mean something like
$$
\vec F\left( \vec v, \hat r, \frac{Mm}{r^2} \right)
$$
Terms like $\vec v \cdot \hat r$ and $\vec v \times \hat r$ are ruled out by the observation that $\vec F$ is parallel to $\hat r$. So, we have
$$
\vec F = -G' \frac{v}{c} \frac{Mm}{r^2} \hat r
$$
where $G'$ isn't necessarily the $G$ measured in the Cavendish experiment (constraints will be derived). Since the sun and the orbiting object might be moving in our Copernican/Newtonian "universe", for the solar system this becomes
$$
\vec F = -G' \frac{\left| \vec v + \vec v_{orb} \right|}{c} \frac{Mm}{r^2} \hat r
$$
There are two simplifying limits, where the sun's velocity is 0 or much greater than the planet's orbital velocity.
Case 1: Sun's "universe" velocity is zero
Then, for an orbiting planet, the acceleration is
$$
\vec F = - \frac{G'Mm}{c} \frac{|\vec v|}{r^2} \hat r
$$
This is still a centripetal force (with no torque, $\vec r \times \vec F = 0$). So, it works out that angular momentum $L$ is conserved. So, the effective one dimensional DE is, since $L = m r^2 \omega$ and $v = r \omega$,
$$
\frac{d^2 r}{dt^2} = - \frac{G'M}{c} \frac{r \omega}{r^2} + \frac{L^2}{mr^3} = \left(L^2 - \frac{G'ML}{mc} \right) \frac{1}{r^3}
$$
This is directly integrable! Defining
$$
A = L^2 - \frac{G'ML}{mc},
$$
the solution (starting from a point where dr/dt = 0) is
$$
r(t) = r_0 \sqrt{1 + \frac{At^2}{r_0^4}}
$$
Three subclasses:
Subcase 1: $L > G'M/mc$ so that $A > 0$
This describes an orbit spiraling away to infinity, horribly contradicting observation.
Subcase 2: $L < G'M/mc$ so that $A < 0$
This describes an orbit spiraling in. Also bad.
Subcase 3: $L = G'M/mc$ so that $A = 0$
Fine, a falling straight trajectory.
So, this horribly fails.
Case 2: The Sun's "universe" velocity is large
The orbital velocity is much less than the speed of light, but the sun's might not beso expanding in series
$$
\frac{\left| \vec v_{sun} + \vec v_{orb} \right|}{c}
\approx \frac{v_{sun}}{c} \left(1 + \frac{\hat v_{sun} \cdot \vec v_{orb}}{v_{sun}} \right)
$$
This leads to two terms, the first of which is just Newton's laws (absorbing $G'v_{sun}/c$ into $G$) and the second of which is a perturbation:
$$
\vec F = - \left(1 + \frac{\hat v_{sun} \cdot \vec v_{orb}}{v_{sun}} \right) \frac{GMm}{r^2}
$$
I don't intend on solving this more complicated equation, but I'll point out that if you did so (analytically or numerically), you could use observational data to put constraints on $v_{sun}$, which I believe would be severe.
Edit: This one sneaks through to lowest order if you assume that the sun's motion is normal to the solar system's plane; you'd have to look at nasty second order terms.
