An expanding universe doesn't necessarily have the property that any two points accelerate away from each other. It is sufficient that any two "comoving" points move away from each other at a constant velocity. You can think of "comoving" points as points that are at rest with respect to the cosmic microwave background radiation, i.e. points from which the universe and all physical prameters such as the previously mentioned cosmic microwave bacground appears to be the same in all directions.
It is the fact that "comoving" points get further and further apart from each other as time goes on that allows one to say a universe is expanding.
If one accepts the standard cosmological models, in our universe at the current point in time, any two points do accelerate away from each other. However, it is possible for the rate of expansion to be constant, accelerating, or deaccelerating - this is measured by a cosmological parameter q', the deceleration parameter. The sign of this parameter, which can change as the universe evolves, determines the sign of the acceleration between two neighboring points.