Does the fuel gauge on the dash have a fuse?

No, it does not. There is only a variable resistance between the ECM and ground that it uses to calculate the fuel level and send it via the data link to the dash. All of the gauges in the cluster get power from the same fuse. A common issue with those clusters is for the stepper motors to fail with bad solder joints being a close second. If it has stopped working the easiest way to tell if it's the stepper motor is to watch the gauges when you first turn the key on, do the gauges move but not the fuel level? It's the gauge. Another way is to use a scan tool to do a gauge test and see if it reacts to that.

You don't need to replace the entire thing you take it apart and replace just the stepper motor. Or send it out for repair. I would go and look at the jeffescortlx YouTube channel, he repairs a lot of these and shows the stepper replacement multiple times on different GM clusters. You need a small soldering iron and some patience.