A spacecraft in a circular LEO orbits at ~7.7km/s. If it performs a retrograde burn and cancels this velocity (within the Earth reference frame) it will start falling towards the Earth center along a straight line and hit the Earth's surface.
A spacecraft in a circular orbit around the Sun at 1Au radius would orbit at ~29.78 km/s. If it performs a retrograde burn and cancels this velocity (within the Sun reference frame) it will start falling towards the Sun center along a straight line and, unless Venus or Mercury would happen to perturb its trajectory, will burn after getting close enough to the Sun's "surface".
The Solar System is traveling at an average speed of 230 km/s within its trajectory around the galactic center.
Let's imagine a spacecraft that is launched from Earth in direction opposite to Earth motion around the Sun, and performs a burn to cancel both velocities: a) due to Earth's spin and b) the ~29.78 km/s around the Sun, hence becoming "stationary" in the Solar system.
Straight after this the spacecraft performs another burn in direction opposite to direction of Solar system motion in the Milky Way galaxy (and I'm not sure what this direction is, as this related question remains unanswered) in order to cancel the galactic 230km/s speed, hence becoming "stationary" in the Milky Way galaxy reference frame (I know it's practically not possible yet).
Will the spacecraft in this case start falling towards the Galactic Center?
Will it reach the Galactic Center if happenned to be not perturbed heavily by nearby stars on its way? Or maybe motion in galactic scale is affected by the nearby stars so much that the perturbation effects can not be neglected?