That isn't the way the American system works. I lived there for 22 years and studied the constitution as a side line (very interesting stuff).
The budget is required to originate in the house of representatives (the only such bill specifically called out in the constitution), and it was originally envisaged that the executive (president) would be the weakest of the three branches of government.
The President has no power to make laws (he can invoke an emergency but that is specifically permitted in certain circumstances); he (and the rest of the executive) are supposed to actually execute the laws. He can shout all he wants but the only people he is hurting are the federal workers who have no paycheck in sight (apart from his own reputation but as he is pretty well heeled he can afford to retire anyway).
Only congress has any power to actually make any laws (including the appropriations bill). The president can refuse to sign a bill but if the house and senate vote with a 2/3 majority after that, it will become law without his signature.