It's been a long time since I've read Shaara, so I can't speak to the exact context the Lee character says it in the novel. It may not necessarily have reflected Lee's view of the matter, but that kind of ambivalence in regards July 4th was not uncommon in the late antebellum and Civil War South. To be clear, for this response I am relying heavily on the work of Paul Quigley, Associate Professor of History and Virginia Tech and Director of the Virginia Center for Civil War studies, who has been working on this particular topic for over a decade.

The ambivalence itself derived from the mixed meanings attached to July 4th. On the one hand, July 4th was the independence day of the United States, a nation that the Confederate states were trying to leave. There was a certain awkwardness associated with celebrating an independence day that had greatest relevance as part of the national narrative of a state the white South was trying to leave. For this reason, some more radical southern secessionists would try to downplay the importance of July 4th, arguing that it's true significance was as a breaking away from Britain, and underplaying its role in national unification. On this model, the unifying of the 13 original states was merely a pact of convenience that could be dissolved as readily as the convenience itself. If July 4th represents little more than a business agreement, it hardly seems tremendously important.

Other white southerners, however, emphasized the declarative nature of the holiday as a celebration of national self-determination. It was lost on precisely nobody that the southern states were, in their own way, declaring independence. In this way, they could lay claim to being the true inheritors of the legacy of July 4th and of the Revolution itself. In this sense, Confederates were laying a claim to a "pure" form of American nationalism they saw the North as having corrupted. To some white southerners, it was the North who was betraying the Spirit of '76. If you read the text of South Carolina's Declaration of the Immediate Causes, it specifically appeals to the example of 1776 as precedent for the state's secession.

To many white southerners, in fact, their secession was their way of participating in the wave of revolutions and liberalization sweeping much of the western world in the middle of the 19th century. We are talking about a time in which the memory of 1848 was alive and well, and many literate white southerners believed they were carrying on the same struggle for self-determination as the Poles, Hungarians, and others. That many white southerners would thus claim July 4th for themselves should not be surprising.

The white South's perception of July 4th turned more negative in 1865 and remained so for much of the remainder of the 19th century. By this point, the celebrations called attention to what was essentially an imposed nationalism born of defeat in the Civil War. July 4th in some parts of the South was much more celebrated by black southerners than white, for example, though this would cease to be true by the 20th century. The most extreme example of white southern ambivalence or outright hostility towards is the city of Vicksburg, Mississippi, which fell to Union forces on July 4th, 1863. Vicksburg did not publicly commemorate Independence Day again until 1945.

Readings

Paul Quigley, Shifting Grounds: Nationalism and the American South, 1848 - 1865

Paul Quigley, "Independence Day in the American South, 1848 - 1865"

Andre Fleche, The Revolution of 1861: The American Civil War in the Age of Nationalist Conflict