As above really, if you have an eazibleed (have you met my EMO friend? We call her Eazibleed
).
Bleed the air out of the caliper you've replaced. if it's a new caliper, the piston is already retracted. If fitting a second hand one, make your life easier by retracting the piston off of the car, this also gives you chance to 'feel' the seal/piston for any sticking. Once bleed, check fluid level, pump up the pedal and re-check fluid.
I have used an eazibleed before, but I have a pressure bleed that runs on a compressor which I prefer. If you have a compressor, they are really worth having. If you buy a few more lengths of pipe, you can use it for all sorts of jobs. For example I've used a pipe to hoover up the crap in the bottom of the brake reservoir. Or empty the power steering reservoir, then disconnect the return pipe, wedge my suction pipe in the return pipe (may require a few layers of tape to get it snug) and lock it on suck whilst topping up the reservoir. As I use clear pipe, I can tell when the new fluid makes it through the full system.
No I haven't done an engine oil change this way and I doubt I ever will.