How can Newton's idea of absolute space be reconciled with Galilean relativity?
Galilean relativity is Galileo's observation that the laws of physics are the same in all reference frames. Specifically, he noted that if a scientist and his laboratory move at constant speed in an unvarying straight path, the scientist will have no way of knowing his reference frame is in motion. In other words, unaccelerated motion of an object has meaning only in relation to another object.
Although Newton believed that there is an absolute space and an absolute time, he also believed that humans live in relative space and relative time. He defined absolute space as a mathematical construction existing "without regard to anything external" (in his own words). But he also postulated that all motion takes place in relative space, which we incorrectly perceive to be absolute space.
Newton did not deny Galileo's principle of relativity. He supplemented it by saying that relative motion exists in a theater of absolute space and time. The idea of absolute space is not necessary to explain the laws of Newtonian mechanics. Perhaps Newton's idea of absolute space provided a way for him to reconcile relativity with the existence of an absolute God.
Newton's idea of absolute space simply appeared as an answer to the following question: What is an inertial system? Saying that an inertial system is one with constant velocity relative to another inertial system of course does not answer the question. To avoid such logical weakness in Newton's first law one has, at some point, to assume that there is a frame of reference, called absolute space, that - by definition - is inertial.
On the other hand Galilean relativity consists on transformations among inertial frames of reference and such relations do not forbid an absolute space. In fact, the idea is that we can use Galilean relativity transformations to relate any inertial frame to the absolute space.
Although the concept of absolute space can be removed by Mach's definition of inertial frame (given an isolated particle, there is a reference frame, called inertial, relative to which the particle has constant velocity), there is no contradiction between Galilean relativity and absolute space. Only Special Relativity can definitely rule out the existence of absolute space.