How do I remove the passcode from my watch?

Yes that can be done. Check the instructions in here:

Lock or unlock Apple Watch - Apple Support

Note that if you do that, Apple Pay won't be available any more. Is your wrist detection working correctly? Normally that prompt appears pretty much only when putting the watch on the wrist.

Is your paired iPhone perhaps registered with your employer, with a tighter security profile in place? I mean are you required to use a passcode on your iPhone as well? In that case that profile may govern these settings as well.

You can check this under Settings > General > Device Management