C'est moi:
<i>so the velocity [of an orbiting planet] is due to gravity</i>
No. Not at all. The velocity is just something the planet happens to have had from the time of the formation of the solar system. Gravity acts at right angles to the velocity (for a circular orbit). Therefore, the gravitational force does not change the <b>speed</b> (i.e. the magnitude of the velocity), but it does change the direction of motion. A change in the direction of motion is a change of velocity, and a change of velocity is an acceleration. So gravity is correctly said to accelerate the Earth in its orbit around the sun.
We do not feel the acceleration because we are accelerated by the sun's gravity at the same rate that the rest of the Earth is accelerated. Both the Earth and the people on it are in free fall towards the sun at all times. We don't move closer to the sun, though, because our pre-existing velocity makes us go round. We continually fall towards the sun, but "miss".
Think about throwing a ball horizontally off a high tower. If you drop it (zero horizontal velocity), it simply falls to the ground. If you throw it outwards (some horizontal velocity), it hits the ground some distance away. If you could throw the ball very fast, as it fell towards the ground, the ground would curve away from it (due to the Earth's curvature). The ball would never hit the ground, although it would be continually pulled downwards by gravity. In other words, it would be in orbit, for the same reasons that the Earth orbits the sun.
(Q):
<i>If we take the Earth-Moon system for example, the force of gravity of the orbiting body, the Moon, must also be taken into account as another force acting on the system.</i>
The force of gravity due to the Moon acting on the Earth is equal and opposite to the force of gravity due to the Earth acting on the Moon (see Newton's third law). In fact, the Earth and Moon orbit their common centre of mass. However, the centre of mass is a point inside the Earth, so it is more or less correct to say that the Moon orbits the Earth.
My point was that there is no "outward" force on the Moon. The only force on the Moon, from an inertial point of view, is the centripetal gravitational force from the Earth. Thus, there is no
"balance of forces" if you look only at the Moon, for example.