The purpose of the Declaration of Independence was to make a moral case for dissolving the legal ties between the colonies and Great Britain; once that goal was achieved, the official role of the Declaration was finished.
George Athan Billias, Jacob and Frances Hiatt Professor of History at Clark University summarizes here why, in his view, arguments like Cline’s fail. I don’t wish to re-hash the argument. However, it is interesting to note that the Declaration has often served as crucial bedrock support for the growth of freedom in America. For example, much of the debate over Missouri’s admission to the United States centered upon the tension between the two documents. At issue was the applicability of the Declaration’s principle of equality to practical political questions regarding Congressional power and the extension of slavery to new states. Of course, this dispute was not definitively resolved, as evidenced by the Missouri Compromise and the Dred Scott decision.
Abraham Lincoln’s views on this subject are particularly noteworthy. A manuscript fragment of Lincoln’s thoughts, written (most likely) in the early days of the Civil War, addresses the question in words that play with a figure from Proverbs 25:11: “A word fitly spoken is like apples of gold in a setting of silver.” To Lincoln, the image illuminates the distinction between the picture and the frame. In his reading, the Constitution is the frame that contains the golden ideal — that all men are created equal — advanced by the Declaration.
“The assertion of that principle, at that time [of the Revolution], was the word, ‘ fitly spoken’ which has proved an ‘apple of gold’ to us. The Union, and the Constitution, are the picture of silver, subsequently framed around it. The picture was made, not to conceal, or destroy the apple; but to adorn, and preserve it. The picture was made for the apple — not the apple for the picture.” (These words are cited and discussed here). This concept appears again in Lincoln’s Gettysburg Address, wherein he rhetorically asks whether a nation “conceived in liberty and dedicated to the proposition that all men are created equal” can endure.
Accordingly, those who would advance a republic built entirely upon secular grounds argue against Lincoln, seeing the Constitutional picture frame as being designed to conceal or destroy the golden apple of the Declaration and the equality it upholds for all.