The Schwarzschild radius is proportional to c^-2, so this would mean the Schwarzschild radius of the Earth would be about 400 billion km, and the Earth would be a black hole.
If you're allowed to change the speed of light, you can also change the gravitational constant G to compensate since the Schwarzschild radius is r = 2GM/c^2, where M is the mass of the object.
Sure, you could change all sorts of things. Doing it likely means changing so much that things end up looking just like they do now, just scaled somehow.