The square symbol on the Wii U gamepad

Yes, regular Wiimotes are supported by the Wii-U.

This page on Nintendo's website says:

Controllers

The Wii U console is capable of supporting two Wii U GamePad controllers, up to four Wii Remote (or Wii Remote Plus) controllers or Wii U Pro Controllers, and Wii accessories such as the Nunchuk, Classic Controller and Wii Balance Board.

To play Wii Only games it would kind of work if you have a sensor bar, but several games require the gamepad either somewhat (it's required for some levels in Super Mario 3D World but might not be required to reach the "final" boss) or entirely (zombiU and Nintendoland are unplayable without one), and the eShop is completely unusable without the gamepad. I really wouldn't recommend it without the gamepad unless you're using it as a Wii replacement.

Also I'm not sure if first-time setup requires the gamepad (not willing to factory reset to test), but the Settings menu requires it, so for sure you won't be able to set up Wi Fi, download system or game patches or access the digital game store ever. It's really not designed to be used without it and while it technically does not need the gamepad for all uses, it's really not a properly functional device without one.