In English one might say "I have fallen for you". Could this be said in Spanish with caer, e.g. "tu me has caido muy bien"?
No, although the meaning of the sentence you have isn't terribly far off from what you want to say, either, so you would probably not get a correction.
When you say that someone te cae bien, you're saying that you find them affable, a good person, etc. If they te cae mal, that means you find them to not be very good natured. (in general, caer bien is a much better translation of the English like in the sense of I like [someone], as gustar has connotations of non-platonic interest when used with people)
To fall for someone, in the sense of crushing on someone or being head-over-heels in love with them, the most common way is to use enamorarse: Me he enamorado de ti, often with an intensifier like perdidamente or more colloquially (in Spain, at least) hasta las trancas or hasta los huesos.