You typically demand some action. "I demand you leave this house at once", "They demanded that the governor resign", etc.
You can demand an object, like "The customer demanded a refund" or "Hitler demanded the Sudentenland". In such a case the person is demanding that the thing named be given to him, so in a sense you're still demanding an action.
The wording in the example is odd. As kiamlaluno says, a more common wording would be, "He is demanding my father pay him $600" or "He is demanding that my father pay him $600." In either case, he is demanding an action. Saying, "my father to pay him" is a little odd because we don't normally put an infinitive in there.
But in any case, it's still quite common to demand an action.