The tailgate on my 04 Envoy XUV will not open.

Take your vehicle to a qualified repair shop ! You have not the knowledge or testing tools to even begin to diagnose this .

check the fuse/relay. if it is neither of these check the switch with a multimeter. If you can hear the motor trying to move the glass when you try the switch it may be the regulator needs attention. It may be the motor needs to be replaced.

check fuse and relay along with the switch first then the regulator and motor afterwards. Replacing the small electronics is obviously easier than having to open up the tailgate to check the mechanics of the winedow.

The tailgate assembly requires regular lubrication and adjustment in order to keep functioning properly. If it has stopped functioning, follow the manufacturer's recommendations for periodic lubrication and adjustment.

Dealer or a junk yard

this is the exploded view of tailgate https://www.gmpartscenter.net/v-2004-gmc-envoy-xuv--sle--4-2l-l6-gas/body--tailgate this is removal and working on an xuv https://www.youtube.com/watch?v=MLQLAD8xOPk

Is this the XUV version? There is a known problem with all Trailblazers and Envoys with the wiring harness at the top of the gate. Pull that little rubber boot off and look at the wires in there. Are their chaffing wires in there (rubbing against the metal door, which will remove the plastic covering on the wires)? If so, put some black electrical tape (99 cents) around the exposed wire and this should fix your problem. Also, check the rear fuse box located under the rear drivers side seat (just fold the bottom part down. There is a fuse in there that controls the tailgate, check this fuse, chart will be either on the bottom of the cover or in your owners manual as to which fuse.

If both of these check out to be good, could be the lift gate control module

do you have the tailgate with the window in the upper half and does the window open?

pull the plastic off of the tailgate and make sure the small wire looking controls are still hooked up. sometimes they will pop out after alot of use. other thing is make sure the rear actuator is actually working (it could be bad)

I also have a 2004 GMC XUV. My gate would go down but not swing open. In trying to fix it, the gate then refused to go to the flat position.I have removed the cover plate on the gate and found the one of the solenoids loose from its mounting. Resetting the computer by disconnecting both battery leads and holding them together for ten seconds worked to restore function to the gate going flat, but the swing still does not work. I believe it is a bad solenoid or motor for the bottem left latch pin.

the actuators have a white stem that sticks out of the bottom of them. if you push up on the ones on the driver side it will open the back hatch for you.