No, a unit test is a stand alone bit of code that sets up the environment, runs a bit of code (the unit under test) and checks if the code worker as expected. This does tend to use assertions.

The idea of using assertions, is to put the assertions inside the 'unit under test' so they are checked every time the code is run (sometimes with a way to disable for performance). Then you can run the code normally, and don't need to write a separate bit of code, that has to set up a proper environment (usually with a lot of 'mock' objects).

This style can probably test less, but still works well for 'design by contract'. You can confirm the caller stuck to any requirements of the input, and you can confirm the code stuck to any post-conditions on it's results.