Dante said he would kill him to stop him, but I don't believe he meant he would do so in cold blood, rather that Vergil was forcing him to - all the signs indicate Dante did not mean exactly what he said... his running toward him at the end, reaching out, even shedding tears after.
I am aware of that. I know he didn't mean it in a cold blooded way.
I think he meant he wanted to stop Vergil from unleashing and gaining Sparda's power - Dante apparently believes that the power is too much for Vergil (or anyone), and that having it wouldn't do him or the world any good. In DMC4 Dante also states that he doesn't want demonic weapons and power floating about in the wrong hands, and would rather keep guard over them himself, because he, Dante, does not want to use them for the purposes of conquering or gaining anything.
But that's the thing though, Vergil told Dante indirectly to get back to human world. And to access Sparda's full power you need both amulets. And at the time Dante had his amulet, and Vergil seemed to gave up the idea of fighting for Dante's amulet.
At the end, there's that scene where Dante tries to grab Vergil and Vergil stops him by slashing his hand with Yamato. This implies to me that Vergil either wanted Dante to back off and not save him (which was why he edged to the cliff so that he fall deeper into Hell and knew Dante wouldn't follow him) and from Dante's final reaction to it, it seems that he accepts that that was Vergil's decision and there was nothing he could (or should) do about it. As long as Vergil couldn't use the power (and had lost the Force Edge), there was no need to stop him doing what he wanted to do with his own life.
(Although some people have suggested Dante ran forward at the end to get Vergil's amulet - gaining all of Sparda's potential power for himself, the other amulet and the sword and so having the means to control the entrance to Hell - but I don't think that could be the case since Dante isn't so much preoccupied with the acquisition of Sparda's stuff than he is with making sure nobody else uses it.)
I figure - they fought, Vergil lost, and Dante tried to stop him falling out of pure instinctual reaction, since Vergil was no longer a threat, and he still cared about him. Rather than him wanting to stop him from entering Hell on his own, I reckon he just wanted Vergil disarmed. From their brief conversation before the final battle, Dante seems to think that Vergil is being silly and that all that power was ultimately meaningless - he was trying to convince him to give up his quest for power and return to normality, and he was serious about it because he did it without any banter or playing around. He didn't leave him behind because, I guess, at the end he was trying to help Vergil.
The first fight between Vergil and Dante in the demon place was for the power of Sparda. And Dante beat Vergil. The cutscene fight at end though seemed to be Vergil telling Dante "I am staying in demon world,you go back to human world", and Dante replied "I won't let you do that even if i end up killing you in the process".