@ptillisch Thank you for your suggestion. I ran the application using Terminal and captured the log output. It's very long, but I've highlighted and posted a few areas that might be of interest for diagnosis. I can post the full log if you'd like.
I also entered the URL into my web-browser and the tar.bz2 file download without issue. It doesn't look like the destination is blocked by our firewall.
Log output
Board watcher
daemon INFO INFO[0009] Updating index url="https://downloads.arduino.cc/packages/package_index.tar.bz2"
root INFO core-client-provider [platform-index]
root INFO core-client-provider [library-index]
Update for version 2.0.1 is not available (latest version: 2.0.1, downgrade is allowed).
2022-10-28 10:04:29.163 Arduino IDE Helper (Renderer)[18867:5762534] CoreText note: Client requested name ".NewYork-Regular", it will get Times-Roman rather than the intended font. All system UI font access should be through proper APIs such as CTFontCreateUIFontForLanguage() or +[NSFont systemFontOfSize:].
2022-10-28 10:04:29.163 Arduino IDE Helper (Renderer)[18867:5762534] CoreText note: Set a breakpoint on CTFontLogSystemFontNameRequest to debug.
root ERROR ---- !!!SLOW!!! DURATION: search took 1149.759 ms. Args: [{"type":"Updatable"}] ----
root INFO ---- DURATION: search took 77.559 ms. Args: [{"query":"Arduino_BuiltIn"}] ----
root INFO >>> Starting boards package installation...
discovery-log INFO stop
discovery-log INFO >>> Stopping boards watcher...
discovery-log INFO Canceling boards watcher...
discovery-log INFO disposing requesting cancel
discovery-log INFO disposing canceled
discovery-log INFO error received
discovery-log INFO cancel error received!
discovery-log INFO stop received event: either end or cancel
discovery-log INFO stop stopped
discovery-log INFO received end
daemon INFO INFO[0011] boards watcher stream closed
Discovery log
Part 1
discovery-log INFO start
discovery-log INFO start new deferred
root ERROR Request install failed with error: 13 INTERNAL: Server responded with: 403 Forbidden
discovery-log INFO start request start watch
discovery-log INFO start requested start watch
discovery-log INFO start resolved watching
daemon INFO INFO[0022] Loading libraries index file index=/Users/smcdaniel/Library/Arduino15/library_index.json
INFO[0022] starting discovery builtin:mdns-discovery process
INFO[0022] starting discovery builtin:serial-discovery process
Part 2
discovery-log INFO {"eventType":"add","port":{"matchingBoardsList":[],"port":{"address":"/dev/cu.Bluetooth-Incoming-Port","label":"/dev/cu.Bluetooth-Incoming-Port","protocol":"serial","protocolLabel":"Serial Port","propertiesMap":[]}},"error":""}
discovery-log WARN Port '/dev/cu.Bluetooth-Incoming-Port Serial Port' was already available. Known boards before override: []
Part 3
discovery-log INFO start
discovery-log INFO start new deferred
root ERROR Request install failed with error: 9 FAILED_PRECONDITION: Library 'Arduino_BuiltIn@1.0.0' not found