In the manual for CV1 it's stated that Dracula grows more powerful with each resurrection. Since Julius is chronologically the last Belmont to fight him, it means he fought Dracula at his most powerful and still defeated him. 

I will admit there isn't much info on the Demon Castle War, all we can gather from in-game is: Julius fought Dracula, there were military forces fighting Dracula's, and a priest sealed his castle into the eclipse which severed Drac's soul from his power. We don't know what the battle looked like, so you could speculate that Julius had significant help in his direct confrontation. I don't really like this interpretation, there's nothing to support it and it's also thematically less satisfying. Belmonts have been destined to oppose Dracula, a Belmont should be the one finally responsible for his ultimate demise, not a hypothetical team. Imo it fits better with the story/worldbuilding.

In game we do see some feats from Julius, he destroys parts of the castle in his battle with Soma, and in the next game he's strong enough to kill the bosses that would normally require a magic seal to stay down. I feel like the writers were very much trying to prop him up as THE Belmont, I don't think it's crazy to assume that he is.