If you read through Washington's address it becomes apparent why the Senators declined to sit while the address was being read. He counsels against the sort of animosity and division common today and advises that we distrust anyone who would try to create them. In fact much of what he says suggests that he would be opposed to the state oriented flavor in the campaigns of current candidates for public office. Here's some relevant text:
The unity of Government, which constitutes you one people, is also now dear to you. It is justly so; for it is a main pillar in the edifice of your real independence, the support of your tranquillity at home, your peace abroad; of your safety; of your prosperity; of that very Liberty, which you so highly prize. But as it is easy to foresee, that, from different causes and from different quarters, much pains will be taken, many artifices employed, to weaken in your minds the conviction of this truth; as this is the point in your political fortress against which the batteries of internal and external enemies will be most constantly and actively (though often covertly and insidiously) directed, it is of infinite moment, that you should properly estimate the immense value of your national Union to your collective and individual happiness; that you should cherish a cordial, habitual, and immovable attachment to it; accustoming yourselves to think and speak of it as of the Palladium of your political safety and prosperity; watching for its preservation with jealous anxiety; discountenancing whatever may suggest even a suspicion, that it can in any event be abandoned; and indignantly frowning upon the first dawning of every attempt to alienate any portion of our country from the rest, or to enfeeble the sacred ties which now link together the various parts.
For this you have every inducement of sympathy and interest. Citizens, by birth or choice, of a common country, that country has a right to concentrate your affections. The name of american, which belongs to you, in your national capacity, must always exalt the just pride of Patriotism, more than any appellation derived from local discriminations. With slight shades of difference, you have the same religion, manners, habits, and political principles. You have in a common cause fought and triumphed together; the Independence and Liberty you possess are the work of joint counsels, and joint efforts, of common dangers, sufferings, and successes.
But these considerations, however powerfully they address themselves to your sensibility, are greatly outweighed by those, which apply more immediately to your interest. Here every portion of our country finds the most commanding motives for carefully guarding and preserving the Union of the whole.
I do think its worth comparing this comment with Madison's observations in Federalist #10. Madison tells us that divisions are inevitable and are to be expected. Is Washington simply being unrealistic?
