I can't improve on the two answers already posted, but I'd like to offer a slightly different viewpoint and claim that the Earth is indeed accelerating outwards.
An object is only not accelerating if it's following a geodesic, so if you are not following a geodesic you are by definition accelerating. The surface of the Earth is not following a geodesic, because an object following a geodesic would (from our perspective) accelerate towards the core, and therefore the surface of the Earth must be accelerating outwards.
We tend to intuitively think of accelerating as accelerating away from some spacetime frame, and indeed in SR this is a reasonable way to think of things. However it's also possible to think of the frame accelerating away from us, and in GR the two are equivalent. In an SR approach you'd argue that the surface of the Earth can't be accelerating outwards because if it was the volume of the Earth would be changing. However when you stand on the surface of the Earth the geodesics passing through you are curving away from your trajectory. In this sense spacetime is accelerating away from you.