Is it possible to set up and watch a game between just computer AI opponents?

This is what I found:

CIVfanatics

It seems to be possible by downloading the CIV 5 SDK called Fire Tune. I'm sorry for not being able to provide you with more information.

I'm sorry no the game requires at least 1 human player in order to begin.

Yes, it is possible. There needs to be a human player to start the game, but all you have to do is step out of the way. One way to achieve it is to use the in-game-editor mod to delete your initial city and hide some inaccessible units. Then, all is left to do is to end turns and observe.

For an example, see the Civilization V AI only series by reddit user thenyanmaster:

• Link to setup and first installment
• Link to list of all published installments