The actual ordinance of nullification that you mention does not really set out Calhoun’s interpretation of the Constitution. It is simply an ordinance stating that South Carolina feels the tariff is unconstitutional and setting out all the ways in which the state will not comply with the tariff.
It seems to me that you are really asking what each side in this conflict over states’ rights believed. Calhoun’s belief, stated elsewhere much more than in this ordinance, was that the states had created the Union and the states could withdraw from it at any time. Jackson’s statement, by contrast, does directly address the issue of states’ rights. However, Jackson does not make an argument based on the words of the Constitution or the manner in which it was ratified as Calhoun does. His argument is based more on what he sees as common sense. The gist of this part of his proclamation is that it doesn’t make any sense to allow states to nullify federal laws or to secede from the Union. He is saying that such a doctrine would lead the country to break up.
The gist of their respective arguments, then, is that Calhoun says that the ratification of the Constitution shows that it was a contract between the states and that states can therefore nullify unconstitutional laws or leave the Union if they wish. Jackson says that the Union cannot last if that is true. Therefore, he argues, Calhoun must be wrong.