No, the Wii U GamePad cannot function without the Wii U console.
Why?
It is a controller, not a standalone device—no operating system, no independent processing, no battery for standalone use (it charges via console cradle/AC adapter).
It communicates via a proprietary wireless link to the console for all data, video, audio, and power.
What You Can Do with GamePad + Console
Play games and navigate menus.
Use its screen for Off-TV Play (play entirely on GamePad).
Run apps like Internet Browser, Nintendo TVii (discontinued), Miiverse (discontinued).
The GamePad is an integral part of the Wii U system and has zero independent functionality.