Was there a naturalization process for those born in the Confederacy after the Civil War ended? Or were they automatically given United States citizenship?

(The true answer to the question is mostly in the last two paragraphs. The previous paragraphs are to give context and allow for a greater understanding, but you are free to skip them if you so wish.)

The question of Confederate citizenship is tricky to say the least. The Lincoln administration never deviated from its position that the Confederacy never existed as a legal entity, even if it existed as a matter of fact. Consequently, there never was a Confederate citizenship that was recognized by the US, and the territories that made the CSA were considered to have always remained part of the Union, just "hijacked" by hostile forces. Of course, Lincoln had to recognize that there existed a Confederacy and that it had ruled long swathes of land for four long years.

Complicating the matters is how the conception of citizenship was different back then. While nowadays citizenship is closely aligned with our cultural identities, back then citizenship was seen not as forming part of the nation, but a status that allowed you to play a part in its political life. Citizenship conferred rights and responsibilities upon the individuals that had it, and here's the key to understanding their conception of it: not everyone had citizenship, even before the war, and there were differences between national and state citizenship.

Of course, Black Americans did not have citizenship. In the South they didn't even have rights of any kind. The infamous Dred Scott decision expressed that Black Americans could not be United States citizens, and, in a direct contradiction of the Constitution, that even if an African-American was granted citizenship by a state, they would not be entitled to privileges or protection in other states. Chief Justice Taney said this because some Northern states, such as Massachusetts, had given Blacks citizenship.

Here's where we find another key element: the divisions of rights. In their arguments, Republicans divided the rights of a person in natural, civil and political. Natural rights are self-evident: life, liberty, property, the pursuit of happiness. Civil rights are needed to defend those: suing in court, presenting evidence and giving testimony, moving wherever you want. Political rights include voting, running for office, more briefly, taking part in any political process.

Republicans pushed for Blacks to gain natural and civil rights, but not political, except, of course, for the Radicals among them. They pointed out that White women and children enjoyed natural and civil rights, but not political. In fact, this meant that women, children, poor people, and others who could not participate in politics were citizens, but did not enjoy the privileges of citizenship. That seems like such a paradox when seen from our modern perspective. The important thing to understand was that even if you were a citizen, you were not automatically granted all rights related to that.

After the war, the conception of citizenship started to change. For one, citizenship was not something that was granted, but you were a citizen because you were born in the United States. Thanks to that, you could now enjoy the political rights related to your status. This meant enfranchisement for Black Americans (women and children, though they were citizen, were still denied their political rights). But what about Confederates?

The terms of surrender for Lee at Appomattox, copied by other surrendering Confederate armies, protected the common soldier from being prosecuted for treason or bothered by US authority as long as they complied with the laws. President Johnson would later issue a blanket pardon that restored the privileges of citizenship as long as they pledged loyalty to the US, except for 14 categories who needed to individually apply for a pardon, and included members of the Confederate government and military officers.

So, if a baby was born in the Confederacy, legally it was considered that he was born in US territory, because, remember, the legal position of the government is that the Confederacy had never stopped to be US territory. As a result, they would be granted US citizenship and all the rights that came with it (if applicable) following the terms of the 14th amendment. For common soldiers and civilians, they still remained US citizens (again, the official position was that they never stopped being US citizens in the first place) and would be able to exercise that citizenship after Johnson's pardon. For people who took a part in the rebellion, they had to apply individually, but Johnson was known for granting these pardons liberally.

To summarize, Confederates remained citizens of the US during the rebellion, and babies born in the Confederacy would be granted automatic US citizenship per the terms of the 14th amendment. They were not stripped of their citizenship, just their right to exercise the political rights related to it, and the only kind of "naturalization process" that ever arose was the individual applications for pardons certain individuals had to submit.

Sources: McPherson's The Battle Cry of Freedom; Foner's Free Soil, Free Labor, Free Men and Reconstruction: America's Unfinished Revolution; Oakes' Freedom National: The Destruction of Slavery in the United States. Daniel Walker Howe's What Hath God Wrought gives greater context relating to the conception of citizenship and the rights of Black Americans previous to the war.