Evidence that stationary masses in space actually attract each other I'm finding it rather difficult to find experimental evidence that two stationary masses in space (unaffected by external massive bodies or gravities) actually attract one another. For moving masses, this is abundantly clear (planets, asteroids, etc.), but who has actually tried to measure forces of attraction between objects stationary in space with respect to the Sun, and has found through experimentation that the hypothesis of gravity being proportional to motionless masses is true?
I'm aware of the Cavendish experiment, however, this experiment is not what I'm looking for because the two balls are moving with the Earth, so they are not completely without motion with respect to the Sun. Relative to each other, the balls are stationary, but I am looking for an experiment conducted where there is no motion in the massive objects relative to the Sun.
 A: 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.
A: Your concern seems to be that the law of gravity might depend on whether the two bodies are moving relative to a third, external reference frame...
For example, you could do the Cavendish experiment in a lab in a University basement. Then you could load all the apparatus on a train and repeat the experiment while the train was rolling along (assuming no vibration and bends and jolts). You wonder if the results might be different, am I right?
I think you'd find there would be no difference and here's why:
Take a brick and count up all the protons and neutrons in it. Using the mass of the particles from particle physics, calculate the mass of the brick. Keep it stationary on the Earth and weigh it. This gives you the gravitational force due to the mass of the Earth.
Now watch the Moon as it orbits the Earth. Calculate the gravitational force that is required for the Moon to orbit as it does.
Taking account of the fact that the Moon is further away than the brick, check if the two results for the strength of the gravitational field are the same.
If they are, then the gravitational strength doesn't care if the object being attracted is moving (Moon) or stationary (brick).
A: The first measurement of the gravitational constant was done by  Henry Cavendish in a lab, in which the gravitational force between two lead balls was measured. They weren't moving. http://en.wikipedia.org/wiki/Cavendish_experiment
A: The concept of a body being "completely without motion in space" is thoroughly discredited, and you really should take the other responders seriously on this point.
According to Einstein's General Theory of Relativity, the nearest we can come to this state is by being in gravitational free fall. But now the Solar System provides the perfect test that you are looking for: every non-artificial body in the Solar System is in gravitational free fall. (OK, there are exceptions: for instance, when a small comet approaches the Sun, it might emit enough gases to measurably change its trajectory. But this is a side issue.)
In short, what you are looking for does not exist. It's like asking whether anybody has travelled to the edge of the world to find out whether the world has an edge or not. "But...but nobody's ever gone there! How can we know it doesn't exist if nobody's ever gone there?"
A: I think Cavendish demonstrated this:
http://en.wikipedia.org/wiki/Cavendish_experiment
