It doesn’t seem that natural, as 水です seems to mean “It is water”, rather than a shorter form of 水を飲みます.
Yes, you can safely say 水です. The implied subject ("it") refer's to "what I want to drink".
Moreover, in Japanese, it's even perfectly natural to say 私は水です if there is enough context! See: Are possessive particles implied in a conversation?
It's not incorrect. It sounds a bit "matter-of-fact" and kind of "abrupt" though.
Also, this largely depends on who's asking in what context. If it's your acquaintance who is about to order drinks for both of you at a restaurant, you could maybe say 「（私は）水で」or「お水を一杯」etc depending on the tone you're going for. If they're asking out of curiosity (i.e. "what will you drink (as opposed to other people)?"), you could answer with「私は水を」, and here 「水です」would sound most natural to me. In an informal context when asked「何を飲む？」by a friend etc, you could also just say「水」, although 「水で」is more standard. Also, if you're in a place where you would be expected to order an alcoholic beverage, 「（私は）お水で結構です」、「水でいいです」 as in "Water is fine, thanks." would also work.
If the person asking is a (somewhat relaxed) waiter or waitress asking for your order,「水をください」「水をお願いします」「水でいいです」「水で」are probably the best options.
If, on the other hand, you were visiting the home of an associate/superior and they (or their wife/husband/son/daughter etc) asked you 「何を飲みますか？」(as if it was natural to give you something to drink), you could maybe say「では、お水を一杯、お願いできますか」or something. This would be more polite than the other ones above.