TL;DR: a close reading of "The Final Problem" says YES, but a broader analysis would tend to reject this evidence and possibly to say NO instead.
Holmes certainly considered Moriarty to be
the most formidable criminal he ever encountered.
From the very first time we see him speak of Moriarty, it is always in tones of awe:
"Aye, there's the genius and the wonder of the thing!" he cried. "The man pervades London, and no one has heard of him. That's what puts him on a pinnacle in the records of crime. [...] He is the Napoleon of crime, Watson. He is the organizer of half that is evil and of nearly all that is undetected in this great city. He is a genius, a philosopher, an abstract thinker. He has a brain of the first order. [...] In three days—that is to say, on Monday next—matters will be ripe, and the Professor, with all the principal members of his gang, will be in the hands of the police. Then will come the greatest criminal trial of the century, the clearing up of over forty mysteries, and the rope for all of them; but if we move at all prematurely, you understand, they may slip out of our hands even at the last moment."
-- "The Final Problem" (emphasis mine)
And after his return from their apparent mutual destruction, he continues to speak of Moriarty with the same respect:
"Well, well, such is fame! But, then, if I remember right, you had not heard the name of Professor James Moriarty, who had one of the great brains of the century. Just give me down my index of biographies from the shelf." [...] "My collection of M's is a fine one," said he. "Moriarty himself is enough to make any letter illustrious, and here is Morgan the poisoner, and Merridew of abominable memory, and Matthews, who knocked out my left canine in the waiting room at Charing Cross, and, finally, here is our friend of tonight."
-- "The Adventure of the Empty House" (emphasis mine)
He also describes Moriarty's right-hand man, Colonel Sebastian Moran, as "The second most dangerous man in London." Presumably Moriarty, then, is whom Holmes considers to be the most dangerous man in London - a high accolade, given how many criminals Holmes has met.
But none of this really answers the question of the nature of Holmes's
personal relationship with Moriarty. Did he consider him to be his "arch-enemy", or simply another, albeit particularly accomplished and dangerous, criminal to catch?
To answer this, we need to look more closely at his words in "The Final Problem":
"I tell you, Watson, in all seriousness, that if I could beat that man, if I could free society of him, I should feel that my own career had reached its summit, and I should be prepared to turn to some more placid line in life. Between ourselves, the recent cases in which I have been of assistance to the royal family of Scandinavia, and to the French republic, have left me in such a position that I could continue to live in the quiet fashion which is most congenial to me, and to concentrate my attention upon my chemical researches. But I could not rest, Watson, I could not sit quiet in my chair, if I thought that such a man as Professor Moriarty were walking the streets of London unchallenged. [...] Again and again he strove to break away, but I as often headed him off. I tell you, my friend, that if a detailed account of that silent contest could be written, it would take its place as the most brilliant bit of thrust-and-parry work in the history of detection. Never have I risen to such a height, and never have I been so hard pressed by an opponent. He cut deep, and yet I just undercut him."
-- "The Final Problem" (emphasis mine)
And the (reported) conversation between himself and Moriarty when they first met face-to-face:
"'It has been a duel between you and me, Mr. Holmes. You hope to place me in the dock. I tell you that I will never stand in the dock. You hope to beat me. I tell you that you will never beat me. If you are clever enough to bring destruction upon me, rest assured that I shall do as much to you.'
"'You have paid me several compliments, Mr. Moriarty,' said I. 'Let me pay you one in return when I say that if I were assured of the former eventuality I would, in the interests of the public, cheerfully accept the latter.'
"'I can promise you the one, but not the other,' he snarled, and so turned his rounded back upon me, and went peering and blinking out of the room.
-- "The Final Problem" (emphasis mine)
All of this suggests something more than mere professional respect for Moriarty's intelligence and skill. Holmes explicitly says that his 'duel' with Moriarty was the greatest height of his career as a detective, and that in order to defeat this single enemy he would gladly accept his own destruction. The above quotes definitely imply a special relationship between the two, which, yes, could definitely be described as their being "arch-enemies" to each other.
However, all of the above is deduced only from close reading. Let's now take a step back and consider the context of "The Final Problem" within the wider Holmes library.
Both in and out of universe, this story was written as a sort of "swan song", a final eulogy of Sherlock Holmes. Both Watson and Conan Doyle thought it was the very end of Holmes when they wrote it, and both had good reason to make it sound as though defeating Moriarty was the greatest achievement of Holmes's career - because it makes a good story, and because it enables the great detective to end on the highest possible note. So the testimony of "The Final Problem" is not necessarily reliable evidence. Even if Holmes didn't consider Moriarty his arch-enemy, his biographers (both in and out of universe) would want to make it sound that way.
So what evidence is there in the other Sherlock Holmes stories? In
The Valley of Fear he plays a prominent background role, and he is again mentioned there as a genius and criminal mastermind, the man behind most organised crime in London, but his personal relationship with Holmes isn't elaborated on as much as in "The Final Problem". The evidence here is inconclusive.
It's worth noting that The Valley of Fear definitely contradicts at least one part of "The Final Problem". In the latter, Watson is unaware of Moriarty's nature as a criminal mastermind until his meeting with his friend mere days before Moriarty's downfall, when Holmes's nets were already almost closed - whereas The Valley of Fear is set long before Holmes is close to catching Moriarty, but Watson is well aware of his suspicions about the man. This again suggests, at least from an in-universe perspective, that Watson is an unreliable narrator in "The Final Problem". (Out of universe, it's more likely that Conan Doyle simply forgot the exact details of "The Final Problem" when writing The Valley of Fear, and neither should be considered more reliable or 'canonical' than the other.)
It's also interesting how few of the stories Moriarty actually appears in. Counting mentions of him as well as actual appearances, he can be found in only seven of the Holmes stories. If he really was Holmes's arch-enemy, then surely a conscientious biographer of Holmes would devote more time to him among sixty stories? This is an (admittedly weak) piece of evidence that perhaps Moriarty should not be considered Holmes's arch-enemy.