Can a ceiling fan be installed on a box with only one black and one white wire?
Yes.

But it depends how your house is wired.  


Some houses have a light switch on the wall, which controls the fan or fan/light together.  In that case there would only be 2 wires - switched hot and neutral.   
Some houses have 2 light switches on the wall, for the fan, and light, separately.  Then you would see 3 wires - fan-switched-hot, light-switched-hot, and a neutral.  
Some houses have no light switch at all, and the fan needs to be switched off at the fan.  They have only 2 wires - always-hot and neutral. 


In the last case, your choices are: use a switch on the fan; use a wireless remote that's part of the fan; or a special control module that mounts in the fan shroud and talks to a remote control (typically installed at a light switch).