...No, it doesn't. All the game says is that Nero is somewhere in the Sparda Bloodlines, and that Nero is Like Dante, but "carries something different." The gam,e gives no indication or even barest implication that Nero was Vergil's son, only the AU Novel that controdicted the game in several important plot spots tried to "say it" and even then shyed off.
And no, neither Reubon, Johnny, NOR Dan ever said "but Nero is obviously Vergil's son", they said "yeah, we told Capcom about ethfans being confused and divided on Nero or what Nero was, and they were susprised it wasn't obvious." The fact that the Director said JUST FOUR MONTHS before release, basically "gome's done, story's set in final state" that it had been ten years after DMC3, DAMNS the idea the Nero is young enough to be Vergil's son; Dante would be 29/30, Nero would be 17 to 19 at youngest.
Ergo, Unless it's a cloning explination, Nero is not viable Vergil's son, and any claim by Capcom that he is makes no damn sense. And no, the twins are not centuries old, they were in their teens in DMC3, stated by Itsuno.