Does gravity push?

Many years ago a friend in a pub suggested that gravity pushes and it's only the existence of a planet beneath our feet that creates a shadow that stops us being pushed away from the planet. If gravity pushes could it explain inflation and, at the same time, the increased expansion of the universe in the later part of cosmological history?

If gravity pushes and it is merely the effect of large masses shielding other masses from the gravity of the rest of the universe that pushes the whole universe apart, inflation could have occurred during a period when there was little or no matter to shield the universe from gravity thus pushing it apart at an extreme rate. Then when matter appeared inflation would have stopped dramatically because the sudden shielding effect.

Now, 14Bn years later the universe appears to be speeding up, thus the invention of dark energy to explain where this expansion is coming from. However, gravity pushing could explain this by the fact that the universe is so expanded by now that it's so far apart the shielding effect is weakened and another period of rapid expansion could be taking place. In other words gravity pushing would only have a cohesive effect on a dense universe but a universe that is sparse (or widely distributed) its shielding effect would be very weak.

Thus we can explain inflation and dark energy, which has always seemed a bit rubbish otherwise. All we need to do is wrap up dark matter in this and we might have a hat trick which could become the new standard model of cosmology.