Wow. Imaginative question, and the stark answer is… “no.”
Although there is stuff in space, it’s so sparse that it’s a vacuum. There’s no “pressure differentials” in the vacuum of space.
We can’t actually “see” stuff moving apart in the universe, but we perceive it by the shifting of light, indicating to us that stuff is expanding.
Originally, it was thought there was a “big bang” event that started the whole thing into motion and that gravity would slow it down, it would collapse, and the whole thing would repeat forever. We think now this is not the case. Some suppose that black holes may be so compressed that inside they change the laws of physics and create alternate big bangs in parallel universes, but the big crunch idea that says things will slow down and collapse is not favored these days.
So yes, it’s expanding, faster and faster – we think (theorize) based on observations, but it’s not due to pressure. Nobody knows what’s causing it, so we came up with an idea of “Dark energy” that nobody knows what it is to explain an explainable event.