Yes, it an be done.  You'd have to unbolt the compressor from the mount, but
it can remain in the vehicle, and the system can remain charged.  You need a
couple of special tools to do the job, however.  Your best bet if you're
going to do it yourself is to obtain a factory service manual.

Before you throw the baby out with the bathwater, check to be sure the
clutch electrical cable is still plugged in to the clutch assembly and
check to see that you possibly didn't break one of the wires. Does the
clutch still engage and/or disengage when operating the A/C?

If the belt was as bad as you say I'm surprised it wasn't screeching like
a banshee long (months) before it broke.