As far as I can figure out, this is actually a problem with Chrome itself. The error posted on the wp-testers list was this:
This is what I get from the Chrome's Java Console when I "Push" the "Enable Gears" button:
"Uncaught TypeError: Object [object HTMLObjectElement] has no method 'getPermission' http://sitename.***/wp-admin/js/wp-gears.js?ver=20080905 (line 21)"
Line 21 in that file is this:
perm = google.gears.factory.getPermission( 'WordPress', 'images/logo.gif' );
The google.gears.factory object DOES have the getPermission method, according to Google's own documentation:
So, not a lot WP can do about it if Chrome is broken.