How can I browse public games in Diablo 3?

You cannot browse public games. If you choose 'public game' in the preferences you will automatically join a public game suiting your difficulty settings chosen by the system.

Just as @dgw says, you cannot browse public games.

When you choose "Public Game" and "Join" it uses the difficulty setting you currently have selected and the activity you currently have selected (Rifts, Greater Rifts, Bounties, etc) to drop you in queue to join another person's public game with those settings.

Once you've selected a difficulty and an activity, the number of players also using those criteria appears at the bottom of the Game Settings window. That's about as close as you can get to "browsing".