He doesn't quite tell us so in as many words but he probably did not and if he did, he worked hard to argue the other way for political purposes. At that point, the content of his heart is not of primary significance.

I must say probably because it's obvious that Lincoln thinks black people and white people are different -he even refers to physical differences- and that they can and should live separately for both parties' good, but issues of racial characteristics and so forth don't seem to have drawn much of Lincoln's attention. He wasn't the type to go around theorizing about brain sizes or otherwise engage with the period race theory as to how and why black people just weren't as good as white people. He engaged with proslavery theory, which is closely allied, but only to pick it apart and oppose it.

The best we have are his political statements, where he is consistently antislavery but has few qualms about appealing to base bigotry in exchange for votes. This makes him pretty normal for his era and occasionally he will go way out ahead of the times, but only for rhetorical effect and then to pull back. It's possible that his lack of engagement with race theory and instrumental attitude toward prejudice enabled him to evolve in the way he's famous for as the war went on. By the end, he's crossed Rubicons unimaginable to most white Americans in the summer of 1860: black men fighting as American soldiers, a durable place for black Americans in a nation that no longer enslaves most of them, a swift and uncompensated end to slavery enacted largely by force, and even the vote for at least those black men who fought for their country or otherwise demonstrated the virtues he wanted in a citizen.

But Lincoln distinguishes between rights he considers natural (we might say human rights) like the right to not be enslaved, political rights (the vote, holding office) and social rights (integration, intermarriage, etc.) From the start Lincoln is convinced of some kind of equality on the first front; he's always antislavery, though his focus is mainly on the threat slavery poses to white republicanism (small-r there), white freedom, and white labor. Moving through toward political rights is where he famously evolves. Neither Lincoln nor many other white Americans, and certainly few to none holding high office during his lifetime, is prepared to endorse social equality.

So is he a white supremacist? Yes. When push comes to shove, Lincoln both tells us and acts as a man committed to ensuring that black Americans are at least informally segregated out of parts of American life. He habitually refers to some black people he employs for years in demeaning ways that he would not use with white inferiors. He believes slavery has damaged the freedpeople and they need whites to take them in hand and teach them how to live independently. He clearly believes that black people are just not quite as good as whites, or behaves indistinguishably from someone who does. In all of these he differs not at all from most white men of his era; even far more radical antislavery men and abolitionists than Lincoln harbor many of the same prejudices.

It's natural to read that and think it doesn't matter. Lincoln was a person of his age and if they were all like that, it's much the same as none of them being that way. It can feel unfair to call him out, like grinding modern day axes with as much justice as expecting him to have an opinion on the space program. This misses at least two vital points that cannot be neglected if we want to understand the history thoroughly. The lesser of the two is that Lincoln was a white supremacist who was much less white supremacist than many of his peers. Excusing him is going too far, but so is imagining him as another John Calhoun or Josiah Nott. There's a complicated range of opinions and modalities for white supremacy and by focusing strictly on the is/is not binary we can tell a simpler story that might be more comfortable to us but ignores historical realities.

The much more important point lost is that there are millions of Americans of Lincoln's era who do believe themselves just as worthy, capable, and talented as any white and deserving of full equality in every form of rights. I refer specifically to black Americans, who fought the struggle against slavery long before any homely guy from Illinois involved himself despite an almost absolute exclusion from political life and most of them being actually enslaved. If we say "everyone" was a white supremacist, we write them and the rest of white America's long list of victims out of the national story. We would, whether we mean to or not, carry on our part in the project that white Americans began long ago of insisting that only they and their interests matter. We would, to use the word, hold white superior to non-whites ourselves.