2004 Envoy Xuv tailgate does not open.

Latch actuator is probably bad an needs replacing . Your best bet , take it to a qualified repair shop. An have it checked .

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.

have you look in drivers hand book to see if there is a fuse for the tailgate actuator

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

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

Do you live in a dusty area? My sister lives in dusty New Mexico and had a malfunctioning XUV tailgate. My reasoning was that something must tell the tailgate not to drop down if the glass is up, and if it can't tell, then it will only open sideways.
So we removed the trim panel, and the tailgate was indeed full of dust. By reaching my arm in, I found two plastic optical sensors which tell the gate whether the glass is up or down; both were coated with dust. After cleaning these the tailgate was back to its normal operation. The two sensors are flat, round - about the size of a nickel or dime...I couldn't see them, just found them by guessing that they must be there. Hope this helps!

Take off the trim panel and see if you can see the servo that controls the locks push rod. It may have been disconeted, or broken, the ends are usually made of plastic. Also you will be able to see if it is even actuating when you press the lock/ unlock. Good luck!

First, on the rear hatch, there is an actuator. On FX, when you close the trunk, it kind of **** the hatch closed all the way. When you press lock on the remote, this same actuator locks it as well.
Now,
1) when you unlock the car from the inside, go back there and press the button to open the hatch door manually, do you here any noises?
2) Close all doors, go back close to the hatch button and using the remote , lock the car and then unlock the car. you should hear a sound from the actuator locking and unlocking the lock on the rear hatch!

If you don't hear this, your actuator is inop.
Next step would be to check the fuses inside the car, on the left bottom driver side, and outside on the battery compartment.
It might be just a fuse.
If all fuses check ok, your actuator is most likely gone! Not ea expensive fix, just time consuming!

NOW, if you can hear the noises, then something is either broken inside the actuator or one of the cables is loose!
Hope that helps.
Also, what year and miles on your fx?

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.