Dumbledore values his morality more than the greater good.

Creating Horcuxes requires one to kill. In Dumbledore's (and others') view this is a terrible moral crime. In Chapter Eighteen of Half-Blood Prince we get the following description:

"All I could find was this, in the introduction to Magick Moste Evile — listen — 'Of the Horcrux, wickedest of magical inventions, we shall not speak nor give direction....' I mean, why mention it, then?" she said impatiently, slamming the old book shut; it let out a ghostly wail.

Later in Chapter Twenty-Three Slughorn describes it as requiring the most evil act:

"But how do you do it?"

"By an act of evil — the supreme act of evil. By commiting murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: He would encase the torn portion —"

And when questioned further by Tom Riddle he reiterates the horribleness of creating Horcruxes:

"Merlin's beard, Tom!" yelped Slughorn. "Seven! Isn't it bad enough to think of killing one person? And in any case... bad enough to divide the soul... but to rip it into seven pieces..."

In Chapter Thirty-Five of Deathly Hallows Dumbledore himself refers to murder as "unspeakable evil":

"You were the seventh Horcrux, Harry, the Horcrux he never meant to make. He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child.

Now one could argue that the greater good would have been served by Dumbledore remaining alive, but it is unlikely that Dumbledore would have sacrificed his morality in order to achieve this. Consider all the other times that Dumbledore had the opportunity to commit an immoral act, an act of unspeakable evil, etc. and he chose not to even though the greater good would have been well served by it. Particularly in Order of the Phoenix he had a dozen of Voldemort's top Death Eaters at his mercy. Had he simply killed them all instead of capturing them, who knows how many innocent lives would have been spared (from the future killings performed by those Death Eaters once they escaped)? If Dumbledore (and others on his side) had always fought to kill, perhaps the entire war could have been averted.

As for why Dumbledore places morality above the greater good, it is likely based on the experiences of his youth. At that time he was friends with Grindelwald and together with him championed the cause of the greater good. But eventually he realized that the greater good sometimes creates fuzzy lines, and one false move can throw you over the precipice of morality from which you might never return. Indeed, one could argue that Dumbledore spent the rest of his life trying to avoid becoming Grindelwald, and therefore he never killed anyone even if the circumstances could justify it.

Note the following exchange in Chapter Thirty-Five of Deathly Hallows:

Dumbledore turned his whole body to face Harry, and tears still sparkled in the brilliantly blue eyes.

"Master of death, Harry, master of Death! Was I better, ultimately, than Voldemort?"

"Of course you were," said Harry. "Of course – how can you ask that? You never killed if you could avoid it!"

"True, true," said Dumbledore, and he was like a child seeking reassurance. "Yet I too sought a way to conquer death, Harry."

"Not the way he did," said Harry. After all his anger at Dumbledore, how odd it was to sit here, beneath the high, vaulted ceiling, and defend Dumbledore from himself. "Hallows, not Horcruxes."

"Hallows," murmured Dumbledore, "not Horcruxes. Precisely."

Here Dumbledore considers the actions of his youth to be almost on par with Voldemort. He is only somewhat comforted when Harry reminds him that he spent the rest of his life avoiding killing. That, in Dumbledore's mind, is his penance, and he would be unlikely to sacrifice that to remain alive.

Later in that conversation Dumbledore tells Harry that he never trusted himself with power after that (my emphasis):

"Years passed. There were rumors about him. They said he had procured a wand of immense power. I, meanwhile, was offered the post of Minister of Magic, not once, but several times. Naturally, I refused. I had learned that I was not to be trusted with power."

"But you’d have been better, much better, than Fudge or Scimgeour!" burst out Harry.

"Would I?" asked Dumbledore heavily. "I am not so sure. I had proven, as a very young man, that power was my weakness and my temptation. It is a curious thing, Harry, but perhaps those who are best suited to power are those who have never sought it. Those who, like you, have leadership thrust upon them, and take up the mantle because they must, and find to their own surprise that they wear it well.

Killing people would be precisely the power that Dumbledore would not want to trust himself with. He might start out with good intentions but end up as another Grindelwald or Voldemort. Any foray into such areas of magic would carry the danger of reigniting the immorality of his youth, something he had worked very hard to repress. Indeed, this can perhaps be likened to the reason Snape gives for why Dumbledore wouldn't let him be the Defense Against the Dark Arts Teacher:

"Not quite," said Snape calmly. "He wouldn't give me the Defense Against the Dark Arts job, you know. Seemed to think it might, ah, bring about a relapse... tempt me into my old ways."

So in short, Dumbledore would have found the notion of killing and creating Horcruxes to be abhorrent, and to go against what he had spent his life working towards.

Additionally, the process of "dying", existing as "less than a spirit, less than the meanest ghost", and then regenerating a body is not exactly a picnic. As Slughorn described it in Chapter Twenty of Half-Blood Prince:

"... few would want it, Tom, very few. Death would be preferable."

He had many reasons he’d never do that.

Though he knew of their existence, Dumbledore wouldn’t create any Horcruxes for several reasons.

Dumbledore planned on his own death.

Dumbledore had been expecting that he would die since before he was cursed by the Horcrux - that he had a curse that would kill him only made his plan more straightforward, and ensured he’d need to die soon. His plans accounted for his death, so staying alive through Horcruxes wouldn’t do anything to contribute to it.

“Well, really, this makes matters much more straightforward.’

Snape looked utterly perplexed. Dumbledore smiled.

‘I refer to the plan Lord Voldemort is revolving around me. His plan to have the poor Malfoy boy murder me.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

He also wouldn’t need to stay alive to advise others, as he was able to give Snape instructions through his portrait, Harry had all the information Dumbledore wanted him to, and few others were involved - the Order wasn’t involved at all.

“And the scene shifted. Now, Harry saw Snape talking to the portrait of Dumbledore behind his desk.

‘You will have to give Voldemort the correct date of Harry’s departure from his aunt and uncle’s,’ said Dumbledore.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

Therefore, Dumbledore staying alive through use of Horcruxes wouldn’t give him any practical advantage over the plans he already set in motion to defeat the Dark Lord.

Horcruxes were too Dark for Dumbledore.

Additionally, Dumbledore would be unwilling to use magic as Dark as creating a Horcrux. As Dumbledore himself explains in his notes for “The Warlock’s Hairy Heart”, splitting the soul is itself a Dark act and can have serious consequences.

“The resemblance of this action to the creation of a Horcrux has been noted by many writers. Although Beedle’s hero is not seeking to avoid death, he is dividing what was clearly not meant to be divided — body and heart, rather than soul — and in doing so, he is falling foul of the first of Adalbert Waffling’s Fundamental Laws of Magic:

Tamper with the deepest mysteries — the source of life, the essence of self — only if prepared for consequences of the most extreme and dangerous kind.”
_{^{- The Tales of Beedle the Bard}}

Additionally, it requires a murder.

‘By an act of evil – the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: he would encase the torn portion –”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

Murders are, in essence, neither justified, necessary nor have good intentions - Dumbledore implies a mercy killing wouldn’t harm Snape’s soul.

“If you don’t mind dying,’ said Snape roughly, ‘why not let Draco do it?’

‘That boy’s soul is not yet so damaged,’ said Dumbledore. ‘I would not have it ripped apart on my account.’

‘And my soul, Dumbledore? Mine?’

‘You alone know whether it will harm your soul to help an old man avoid pain and humiliation,’ said Dumbledore.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

So, Dumbledore wouldn’t consider it morally justified.

Actually had he resorted to dark magic in that way, not only would be doing the exact kind of thing that he was fighting the enemy for doing. and condemning himself to a fate that was literally worse than death, but he wouldn't have been any use anyway because having a horcrux would keep him alive but it wouldn't keep him healthy. He would have been flat on his back in agony from the magic poison, probably for the rest of the series. Consider how long it took for Voldemort to recover from his first defeat.

Because such an act is inherently evil and would make him no better than what he's fighting against - the heroes have to make sure they don't become monsters themselves in the process of fighting monsters.

Also, Horcruxes damage your soul and prevent you from moving on to the afterlife, though the soul can be repaired it's extremely difficult and can kill you in the process, considering how deeply Dumbledore wants to see his dead family again, I don't think he'd risk not being able to move on to the afterlife.

There's also the small matter that killing someone with the killing curse requires that one genuinely wants to kill the victim, I can't see Dumbledore genuinely wanting to kill someone, especially not if it's just a cold murder to create a Horcrux and not necessary to defend someone.

Dumbledore with historical baggage

His family history already consisted of a significant amount of tragedy and controversy. This included the death of muggle children and then later his mother and then sister, Arianna. The loss and drama surrounding the events listed here had a substantial effect on Dumbledore's ethos and decision-making. Even though he manipulated Harry into becoming an evil-McGuffin-seeking, dark-wizard-destroying archetypal hero, that doesn't mean that he would want to personally kill others to purposefully split his soul. He knows what consequences can result when magic is abused, misguided, and perverted to gain an advantage in power or even achieve trans-mortality.

As an example, he's in favor of (or at least complicit with) Nicolas Flammel's decision to destroy the Stone that granted him (Flammel) such long life.

He understands that the Mirror of Erised does not represent reality, neither past nor future. The promises are an illusion.

Additionally, he'd prefer, with enough trust in and proof from Harry and Hermione, to spare Sirius' life.

His insistence that, even though Harry and Voldemort share many traits, that their differences are what most importantly characterize Harry, is pivotal to Harry resisting the temptation of evil and evil acts, as well.

Corruption of the soul

Technically Dumbledore has the prowess, likely, to kill and fashion horcruxes. The implementation details would not be of great difficulty or much of a hindrance. The temptation of doing so would be to want to do good. But the fracturing of his soul could possess him to revert to his earlier years of power-seeking at whatever cost; or worse, transform him into the very thing he wishes otherwise to fight: someone like Grindlewald or Voldemort.

As such a powerful and knowledgeable wizard he is, this could be disastrous for both the wizarding and muggle communities. He has enough sensitivity and self-awareness to realize that he can't trust himself with the powers of dark magic, and avoids it almost entirely as teacher and headmaster, except when trying to undo it (by destroying the ring horcrux, etc.) and research it (by tactically hiring Slughorn).

Overarching belief system

“Dark times lie ahead of us, and there will be a time when we must choose between what is easy and what is right.”

Central to his core philosophy, an amalgamation of thoughts and ideas experientially gained from losing loved ones, chasing strength, and battling the malignant, is a conviction that just conduct overshadows expediency. Though he does intend to sacrifice Harry, his choices are governed by what he considers compulsory and integral, not expedient.

Hiding pieces of himself into objects in order to stay quasi-alive and potentially capable to fight after death could be convenient, but would it be "right"? Or would he be resigning himself and surrendering his integrity?