The concept of being an "American" is based, for the most part, not in a place of birth, but in the belief in a certain set of ideals, in my opinion.
Prior to 1860 was there a universal feeling of being a "citizen of the United States"? I find very little reference that people of that time thought of themselves in such a collectivist manner. The more common concept was that the United States was a collection of soverign states bound together in a Republic form of collective government. That the government of a Republic was secondary to the will of the States of the states which made up the Republic. That the United States Government was not the master of the States, but the servant of the States. And that the United States government existed only to serve and protect the States from foriegn influences.
I would say that Lee was an American, so long as there was an original America to believe in, in that he believed in the originalist ideals expressed by the Constitution. He was among the LAST of the regular army officers to resign his commission. But once the Constitutional Government of the Republic collapsed, with the election of Lincoln and the Radical Republican agenda to change the purpose and function of that Government. The displayed willingness of that Government to use the military force to subvert the freewill of the States, thereby bring on the secession of the Southern states, Lee reverted to his origins of birth of being a Virginian.
"As long as the Union
Was faithful to her trust,
Like friends and brethren,
kind were we, and just;
But now, when Northern treachery
Attempts our rights to mar,
We hoist on high the Bonnie Blue flag
That bears a single star."
"The Bonnie Blue Flag" 1861
Lets remember here that "The Bonnie Blue Flag" of this song was the resurrected Flag of the Republic of Texas used while Texas was an independent country unto itself, prior to its admission to the Union of the United States.