When Abraham said "suit" here, he may have meant suit as in high-ranking executive in an organization. If you read past the sentence wich you gave us, it reads as this:
-nor do I deny that such decisions must be binding in any case upon the parties to a suit as to the object of that suit, while they are also entitled to very high respect and consideration in all parallel cases by all other departments of the Government.
as you see here, when he said suit, it may have been representing the South and the North, since it is said: the object of the suit.
Source: http://www.bartleby.com/124/pres31.html, Paragraph 26