Not how it works. The fuel gauge gets 12 volts first and then feeds voltage over to the temp gauge. The fuel gauge will work regardless of whether or not the temp gauge is present. The problem could be one of several things. You will need to check the following things to narrow it down.
1. Check you are getting 12 volts at the red wire connected to the fuel gauge. (I terminal)
2. Check that the speedometer housing has a good connection to ground.
3. Check that the gas tank sending unit has a good connection to ground.
4. Check that the resistance on the pink wire connected to the fuel gauge is appropriate for the amount of fuel in the tank. 10 ohms full, 73 ohms empty. (S terminal)
If that all checks out then it would be a problem with the gauge not grounded to the speedometer housing or the gauge itself.
76 CJ5 Renegade, 304, T150, 3.54 gears, 30x9.5 AT's