Well, it's got all the same notes, right?
People have been updating the timbres of classic works for centuries. Take Bach's Well-Tempered Clavier, for example. That was originally intended to be played on well-tempered clavichords and harpsichords, but nowadays keyboardists perform it on equal-tempered (not the same thing!) harpsichords, pianos, and even synthesizers.
It is possible, of course, to use electronics to approximate the sounds of an acoustic or classical guitar on an electric (piezoelectric pickups appear to be best at this, and then there's always MIDI), but of course it's never going to be exact, and you don't have a resonating chamber to drum on, and you've got closely spaced steel strings instead of widely spaced nylons, but again, those differences are no bigger than those between a clavichord and a piano.
If you want to perform seriously as a classical guitarist, then yes, eventually you would need to purchase a classical guitar (and not just an acoustic guitar like you had, either). But there's absolutely no harm done in grabbing your electric, dialing in a clean tone, and going to town.
I think you'd be doing yourself more justice as a musician to just approach the classic repertoire using the instrument you have, especially if you're interested in continuing to play electric guitar. Don't let equipment get in the way of your own growth as a musician.
Oh, also: Paganini's 5th Caprice (performed by Steve Vai for the film 'Crossroads'), and Yngwie Malmsteen is very fond of performing Bach.