What (Fedora) Linux tool can I use to view my desktop on the same network from my laptop?
You can either use the remote login feature of X and export the display of the desktop to your laptop, but that'd mean the applications from the desktop would have their own windows in the laptop.
If you want to keep them separate, I'd recommend using VNC for remote desktop access. In that way, all the "screen" of the desktop will be "painted" withing the same app in the laptop, the vnc viewer of your choice.
Hope this helps