Nope, TWOxACROSS, if you play on easy automatic, the game will always dump you at another easy automatic run. But with Normal mode you can carry your moves, devil arms, health and runes to the next difficulty setting.
And, btw, the game is pretty "static" in that sense, different from DMC3 where you always can access whatever level you've previously completed in any difficult, DMC1 will make you run all the game again just to get that secret mission ou that blue orb you missed. It's kinda exhaustive, but that's the way the game works. But you always can stick to the difficult you've just finished: the first time i finished hard, i made another run on hard just to complete the secret missions i've not managed to complete in that run, for example.