Is it possible that the cause of gravity (not just planetary) is the force of the universe expanding outward?
Like, if you blow up a balloon it creates pressure from the surface to the center.
No. Vacuum isn't air.
Although there is an interesting theory that it could actually be the cumulative pressure of vacuum energy, the virtual particles constantly forming and disappearing within the quantum foam. The idea is that when it seems like two masses are attracted to each other, it's really that they're shielding each other from a bit of that uniform energy pressure on all sides, and so that creates an imbalance that causes them to be pushed toward each other.
I've also heard another interesting offbeat theory... not sure I remember the details, but it's something to do with the way every particle exists in multiple overlapping quantum states. And the idea is that particles tend toward the most probable state, and statistically, the most probable location for any given particle in an ensemble is toward the center. So that creates a sort of probability pressure that causes masses to try to reach the center of whatever system they belong to.
Of course, these are fringe ideas -- not to the point of being crackpot theories, but alternate scientific proposals that don't have real evidentiary support. The leading view is the general-relativistic one that the presence of energy -- including the energy contained in mass -- alters the geometry of spacetime, so that the paths of objects or light beams in the vicinity of masses are redirected toward them. Or there's the alternate quantum view that masses emit fields of gravitons that create attraction between them, similarly to the way the electromagnetic fields emitted by charged particles create attraction or repulsion between them.
That would suggest that on one side of the Earth gravity would push you down (if the earth is between you and the center of the universe) and on the other side, gravity would fling you out into space.
No, because there's no center of the universe. The universe is effectively infinite and its expansion is uniform.