Apparently, with the default toolbar buttons, you cannot remove the text, which severely limits the number of buttons that can be displayed across the toolbar. You only have a choice of large or small icons. I asked IDM about possible workarounds, and they suggested created a custom toolbar. I tried that, and got satisfactory results (the buttons look similar to those on UE for Windows.). Here's what I did:
(1) Create a new toolbar: Preferences, Toolbars/Menus, Customize Toolbar. There's a place here to add and name a new toolbar.
(2) You can recreate your favorite toolbars using the window with the left and right panes (Toolbar Items and Commands, respectively). My toolbar is rather large, so I didn't want to do this manually. Instead, I edited this XML file ~/Library/Application Support/UltraEdit/Uex.tb0 to copy the toolbar buttons from the existing toolbar to the new, custom one.
(3) Finally, the new buttons are too small, but they can be enlarged: View, View/Lists, Toolbars, Large Icons.

IDM indicated to me that removing text from the toolbar would be available in a future release. This was a topic I discussed with them during the beta, and offered some insight into techniques I had used in personal projects to obtain that result. It seems that it is more an issue with the OSX Unified Toolbar, than it is with UE itself.

I'm using v.2.1.0.9.

I can customize the main toolbar, and I can tell it to show the icons without text. It all works great, until I shut down and restart UE. Every time UE starts up, the toolbar text comes back. Apparently, there is no way to prevent this from happening.

I was able to set up a custom toolbar, and get it to always show just the icons. It's funny how the custom toolbar closely resembles the standard Windows toolbars. It's actually more compact, and you can fit more icons onto it, but it's less Mac-ish. When the toolbar buttons are "toggled", they get a fairly ugly looking dark square put onto them; this could use an aesthetic improvement.

Can you share how you managed to show the icons without text?

I hate to have to create a custom toolbar just to have the same functionality I have on windows version.

VIEW --> VIEW/LISTS --> TOOLBARS --> [ICONS/TEXT] [TEXT ONLY] [ICON ONLY]

Thanks DaveS, it works.