What did Buck vow would never happen to him after he saw Curly die?
Buck vowed that he would never go down in any fight. He knew that once a dog was down, that was the end of him. He would be torn apart ferociously and would die a horrible death.
In the novel, Curly, a friendly Newfoundland breed, tried to befriend a husky dog, the "size of a full-grown wolf." However, the husky wanted nothing to do with Curly. In a shocking burst of brutality, the husky tore Curly's face open from "eye to jaw." Seeing this, thirty or forty fellow huskies rushed to the scene of the conflict. They waited with baited breath and barely suppressed excitement for the moment that Curly would be tumbled off her feet.
When the moment occurred and Curly went down, the group of huskies set in on the hapless dog. Curly never stood a chance against the group. The killing was short and brutal. It impacted Buck in an immense way. He realized that there was no "fair-play" in the wild; the law of the fang and the club would never allow such a luxury. There were only two options as far as Buck could see: to kill or to be killed.
As time progressed, Buck learned the way of his ancestors and immersed himself in the instincts that were his birthright. He took easily to honing the skill of attacking viciously and swiftly. This skill came into use when he fought Spitz for preeminence as the leader of the pack in Chapter 3. In this fight, Buck kept on his feet. He never went down; instead, he went for Spitz's right and left foreleg, breaking both, until Spitz finally went down altogether. Of course, the waiting huskies joined in on the final kill once Spitz lost his footing; the law of the fang demanded it.
The short answer is that he vowed he would never get knocked off his feet. He would see to it that he never went down in a fight. That way, he would be able to stay alive.
He came to this conclusion, as you say, after he saw Curly die. He learned from that that once a dog goes down, everyone else watching is going to go for that dog and kill it.
It is exactly in this way that Spitz will die when fighting Buck later in the book. He will go down and the other dogs will kill him.