Sounds like you are fine, a belt driven fan will always spin as it is driven by engine .

 

That's normal.
"The fan drive viscous clutch (or fan clutch for short) is a silicon fluid filled coupling with a bimetallic coil that is attached to your Jeep's fan. When working properly, the fan clutch will react to the heat discharging from you radiator by spinning faster or slower depending on the cooling needs of your engine. Need I say, it is critical that your Jeep's fan clutch is in proper working order to prevent over heating.
To test whether or not your fan clutch needs to be replaced, simply spin your Jeep's fan by hand while the engine is cold and if it spins freely 5 rotations or more, it will need to be replaced."

 

If the fan clutch is binding, would it cause the pulley to turn when I spin the fan by hand (engine off)?

 

Are you having overheating issues? You never said why you think something is wrong.

 

Oops, good point.
It's because my fan seems too tight. Although I can rotate it with my hand, it won't free-wheel (spin) on it's own. One good push with my hand only causes the fan to free-wheel about 3 inches, at most.
It's also worth noting my engine seems a bit sluggish and I suspect the stiff fan may have something to do with it.

 

I’ll say this much, for me it was fairly obvious that there was something wrong with the fan clutch. First time i’ve dealt with them as well.

Symptoms:

-Fan appeared to be moving very slowly while engine was running. As though I could reach out and stop it with my hand.
-jeep would start to overheat in slow driving/sitting in traffic, but once going 50-70mph temps would be fine
-Fan had almost no resistance when spun by hand

 

I'll say this much, for me it was fairly obvious that there was something wrong with the fan clutch. First time i've dealt with them as well.

Symptoms:

-Fan appeared to be moving very slowly while engine was running. As though I could reach out and stop it with my hand.
-jeep would start to overheat in slow driving/sitting in traffic, but once going 50-70mph temps would be fine
-Fan had almost no resistance when spun by hand

On the flip side, my Jeep was having overheating issues at idle. The fan clutch behaved normally according to all these spin tests, but was still worn enough to cause the problem.

2003 Rubicon