You can watch live games here. Live Chess, Player tab, view.
Watching Live Games

From the Play menu select Live Chess, close the dialog that asks you whether to start new game.
Above the box showing you available challenges, there are tabs Games and Players. Click one of them, then click View (or Watch) next to the game you want to watch.
Note that the games can be sorted by name, rating, etc (click on the column header). You can also navigate through sub-lists of games (the box does not show all 3000+ games) using the controls underneath the list.

aquiredtaste is right, it is available. When in live chess, click on either the games tab or the players tab, you will be gien an option to view the games in progress... I'm watching one right now.

Sorry to be so dense, but I don't have a games tab or a players tab, just a box that says "Play Live Chess." How do you close that?

Sorry to be so dense, but I don't have a games tab or a players tab, just a box that says "Play Live Chess." How do you close that?
Click on it.

Sorry to revive this, but can this only be done via a browser or on a PC?
I also see no such options in my iOS app. I'm only able to create a new live challenge or seek one based on the time control

Took me a while to find this too. I'm not surprised so many people are having a hard time finding the link to watch live games on a computer. Chess.com could perhaps have "Watch" as a separate menu option (like it is on the iOS app). #UserSuggestion

I also took me some minutes to find the menú. Go to Play - Live Chess - PLAY LIVE CHESS! then in the right side of the page appears the current games.
When watching other games I would like for new games to be opened in new tabs so that I can analyze the previous games. Is there a setting for this?

This is still extremely difficult to find in early 2019 so hopefully these instructions are helpful:
1. Click on "Play" (not very intuitive if one wants to merely watch, but yes, click Play):
2. Towards the lower right, click the vague "binocular" icon ("Events" - even though a single game isn't really an event).
3. Type in the full username of the member you want to watch and then click on the live game that shows up. If it's someone you follow (but not a friend), you can find a list of members you're following, of all places, in the settings - https://www.chess.com/settings/following
4. To stop watching the game, hover over the "Game" tab towards the upper right and click the tiny x button.
Watching live games in progress has always been a feature of ICC, and also of , which is owned by Chessbase and therefore costs a bundle unless you get it free by buying CB10 or Fritz 12. I hope that chess.com will add this feature too.