I don’t think WordPress knows to load or run anything from your registerProcess.php file, so what I would do is leave the register_user() function in that file, and then require_once it from functions.php and do the active part of the code – the if .. register_user() inside functions.php where it will be run. (This is probably a bad way to do this, but if it works for you that’s ok)
So functions.php should look like:
require_once('registerProcess.php');
if( isset($_POST['registerSubmit']) ) register_user();
And registerProcess.php should look the same, just remove the if from the last line