What did Buck vow would never happen to him after he saw Curly die?

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.