Obsidian just pings for updates at the moment. There are plans for a toggle to turn auto-update off eventually, but while things are moving fast in beta, this is very low priority.
However, I would consider showing an error dialog like that a bug that should probably be fixed sooner rather than later. Maybe just log the error to the dev console rather than having a popup, at least.
If you agree, I can move this one to the bug reports category