"With a similar sound", is a relative term. Do you mean in terms of tonal qualities, or in the perceived over-all ambiance?
As far as playing goes, listen to Chet Atkins and Roy Clark. They regularly played classical styles on Gretsch and Gibson hollow-body guitars, with outstanding results. So as far as technique goes, certainly it is doable. I play finger-style on electrics all the time, sometimes using Acoustic Modeling, or sometimes just using different EQ settings, mostly because I am too lazy to change guitars onstage. Some of the places I play have small stages, and there is no room for several guitars. I actually like a lot of the John Denver, James Taylor, and Jim Croce stuff I do better on my Gibson ES 135 or Fender Stratocaster better than on my Ovation Balladeer, or Martin.
Recreating the exact sounds is a little more complex. Acoustic guitars depend on many factors for their sound, starting with the way the wood on the sound board, and to a lesser degree, the rest of the body of the guitar, vibrates. Each type of wood has different tonal characteristics, with some being warm and soft, such as mahogany, cherry, and cedar, and others being sharp and pronounced, such as spruce. That's why a Martin sounds different than a Tacoma, or a Seagull. The side and back woods also contribute to the over-all sound, and it is common to mix them, such as using spruce for the soundboard, and tempering it a bit with mahogany on the sides and back. Also, the way the soundboard is braced effects the sound. And lastly, of course, the type and gauge of strings used. Recreating all of these factors digitally is difficult, however, Line 6 has done a fantastic job with their Variax Series of guitars. They do probably as good as it is possible a job in recreating the sounds of various guitars, including classical, and even banjo and sitar. With the flip of a switch, you can go from a Martin D-28, to a hand-made cedar-bodied Classical sound, then with another flip, to a Fender '65 Stratocaster.
So the answer is yes, at least with the right equipment. Of course, there is no need to duplicate the sounds exactly. With the huge range of tonal possibilities of a modern electric guitar using various EQ settings, amp models, and effects, you could create beautiful passages that a straight acoustic could never match. My advice is to press on, don't worry about exact duplications, and make the music your own. That's what it is all about...