Can gravity cause a Big Crunch, if the cosmological principle holds?
No. That's a myth that grew out of the Friedmann equations. Alexander Friedmann was a ballistics instructor on the Austrian front. He modelled the expanding universe as something like a cannonball fired aloft. See John Peacocks’s cosmological physics for an example: “the dynamics of the entire universe are the same as those of a cannonball fired vertically against the Earth’s gravity. Just as the Earth’s gravity defines an escape velocity for projectiles, so a universe that expands sufficiently fast will continue to expand forever. Conversely, for a given rate of expansion there is a critical density that will bring the expansion asymptotically to a halt”.
Image from Norbert Rumiano’s article a cosmological model
It’s totally wrong I'm afraid. Gravity doesn’t stop the universe expanding. A gravitational field is akin to a pressure-gradient in space. It makes matter fall down, but it doesn’t make space fall down. You can tell it’s wrong because Big Bang cosmology is said to be balanced on a knife edge, such that the density of the universe is incredibly close to the critical density. That’s just a re-heat of Einstein’s greatest blunder, wherein a slight increase in density would trigger contraction, and a slight decrease would trigger expansion. It’s also wrong because the 1998 supernova observations didn’t match any of the cannonball trajectories. They didn’t measure the expansion slowing down, they measured it speeding up. Like the ascending light beam speeds up, because the speed of light is spatially variable. Unfortunately people like John Moffat and João Magueijo and other VSL guys said the speed of light in the universe is slowing down, not speeding up. Unfortunately the Friedmann equations feature not just a fixed c, but a curvature k. Even though a gravitational field is a place where the speed of light varies because space is neither homogeneous nor isotropic. Not a place where space is curved.
Say we have a closed universe that follows the cosmological principle.
We have no evidence that the universe is closed, or that the cosmological principle is scientific. I think it's rather like saying I live in a forest, so everybody must see trees.
As a result, mass is distributed equally and, if as far as I know, gravity should have no net effect on the movements of galaxies (as in, there are no peculiar velocities).
The issue isn't with galaxies, it's with space. Space is expanding like the raisin cake analogy, and gravity won't stop that.
However, in the Big Crunch, the Universe stops expanding and gravity causes the scale factor to decrease. How is this possible? Shouldn't gravity have no effect on a homogenous and isotropic universe?
I don't think it is possible. Note that Einstein described a gravitational field as a place where space was neither homogeneous nor isotropic. So a homogeneous isotropic universe is one where there is no gravity.
On that note, if gravity actually caused a collapse, would gravity cause the comoving coordinates of the galaxies to change?
I would say no. I'd also say gravity doesn't cause a collapse.