When one makes a new acquaintance with somebody in person, you may say “it was nice to meet you”, e.g. when you leave. What if you make a new acquaintance over the internet, what do you say when you finish the conversation, business transaction, etc.? You didn’t actually meet the person, you only exchanged some emails. One could say “it was nice to talk/do business/etc. with you” but that doesn’t confer the fact you made a new acquaintance and it was enjoyable experience for you. Does the word meet really work in this case or is something else better?
If you talk with someone on the phone before (or instead of) meeting them in person, it would sound odd to say "nice to meet you." Instead, you would say "It was nice talking with you," and possibly add that you look forward to meeting them (or hope to meet them, at least) in person.
Similarly, an exchange of written correspondence doesn't constitute a meeting either. It would be laughable to use "Nice to meet you" as the close of a letter.
I personally avoid using "meet" to describe an internet encounter, whether in email or chat, even though my fingers often want to type just that. Such an encounter is a kind of meeting, as @kiamlaluno notes in his comment, but it seems so much less than a real meeting that it feels awkward to call it that. Nevertheless, this is the Internet, and what may hold for one medium may not hold for another. I would say that if you have a lengthy, soul-baring chat session with someone via the Web you might well say it was nice to meet that person, since you did exchange more than just a few words. But that would be a figurative usage, and it could apply to a similarly lengthy phone call.
As a final note, people talk about others they have "met online" all the time. So I think this argues for at least considering online encounters as legitimate "meetings" — and if they can be called meetings, I suppose it would not be too surprising to hear "Nice to meet you" in an email. But I think it still involves more than the exchange of a few words.
It depends on the formality of the exchange or, more specifically, the necessary level of formality in language that your relationship requires.
If it's an informal relationship, you can be a little more colourful with your use of the English language and perhaps put quotes around "meet" to indicate that you are using the term metaphorically. You are certainly not the first person to worry about using meet in this way and, I believe, the meaning would be understood.
If the relationship is more formal, then the appropriate phrasing would be dictated by the purpose and content of your exchange. In this instance "It was nice chatting/doing business with you" would be more appropriate.