First of all, the ioquake3 2009 stable release doesn’t work on macOS 10.9 or later. You’ll need to build ioquake3 yourself (see Ioquake3 failed to launch (mac os sera 10.12.6)).
The Mac OS X launcher code and Xcode project live at misc/osxfe/ioquake3fe. I have no idea if it still builds.
There is a newer cross-platform launcher using Qt UI at https://github.com/ioquake/launch. It might be a little more work to compile than the older Cocoa ioquake3fe as you have to install Qt development libraries.
For what it’s worth, the launchers just lets you set a few options like resolution and specify additional command-line arguments. You can use the command-line directly instead of the launcher without any real lost. (Well, the new launcher also lets you install Q3 from a CD and download patch pk3s [pak1-8].pk3 but these aren’t too hard.)
If I remember correctly it can be done in Terminal app like this:
open -a ioquake3 --args +set com_zoneMegs 24 +set r_mode -1 +set r_customWidth 1280 +set r_customHeight 720 +set r_fullscreen 0