Can gravity cause a Big Crunch, if the cosmological principle holds? Say we have a closed universe that follows the cosmological principle. 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).

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?

On that note, if gravity actually caused a collapse, would gravity cause the comoving coordinates of the galaxies to change?
You are thinking about gravity and how it affects the universe as a whole in the wrong way.

The dynamics of the universe are governed by the Friedmann equations, which can be solved to say how the scale factor of a homogeneous, isotropic universe behaves with time.

If there is enough gravitating matter/energy density in the universe then the scale factor may eventually decrease to zero in a finite time. The simplest case is a closed universe with no cosmological constant and a mass density greater than the critical density (which is not the universe we currently inhabit: our universe appears to be (a) flat and (b) expanding at an accelerating rate).

The interpretation of this is that the space between the galaxies is contracting. It is the reversal of what we currently observe in the expanding universe.

In the same way that you do not interpret the universal expansion of a homogeneous, isotropic universe as some sort of repulsion between galaxies; neither would the contraction be interpreted as some sort of attractive force between galaxies. As you rightly point out, in a strictly Newtonian sense, there would be no net gravitational force on any one galaxy and so its co-moving coordinates would be unchanged during the contraction.