Can I just remove rear axle to replace rotor or do I need to remove the whole tire?

In theory this might be possible, but would require some creative repair work. First, remove the brake caliper, to at least get it out of the way. Second, break loose (but do not remove) the bolts that hold the brake disc rotor to the wheel. I imagine this would be similar to loosening and then tightening the lug nuts on a car when changing a flat tire using only the lug wrench. In other words, the wheel must be on the ground to apply enough torque. Then loosen the drive belt (don't have to remove it). Next, you would need to temporarily remove the axle spacers, possibly on both sides, then temporarily re-install the axle again without the spacers, but only on the drive side. Don't put it through the brake side. You might get lucky and only have to remove the spacer on the brake side. This way you wouldn't have to completely remove the axle. This would create a gap to remove the disk rotor. You have to lift the bike off the ground for the axle work. Next, remove the disk rotor bolts and slide the disk rotor out.

Slide in the new brake rotor and lightly tighten the bolts, at least enough to hold it in place. Then reassemble in reverse order including putting the drive belt back on. After the wheel is firmly on the ground, then torque down the rotor bolts. You could do this either by lifting the bike and rotating the wheel, or just moving the bike back and forth on the ground to rotate the wheel. Then you will need to adjust the drive belt to the correct tension. Lastly, put the brake caliper back on and bleed if necessary.

FWIW, there are drawbacks to changing the brake rotor this way, mainly the inability to inspect everything, which you could do if the wheel were off the bike. You couldn't replace bearings or seals or anything like that, without fully removing the wheel.