I would like to isolate a given protein as an intact and functional protein by flow cytometry (without protein unfolding or denaturation), can I do that?
I think you may be seeking FPLC (http://en.wikipedia.org/wiki/Fast_protein_liquid_chromatography). Protein purification can also be accomplished without an FPLC using column chromatography via gravity flow if an FPLC machine is not available. Flow cytometry is very different, and is typically used to separate and measure intact cells, rather than individual proteins. As far as protein purification, I recommend beginning by searching the literature for instances in which the protein has been purified - this would be a good sign that it is generally soluble. If there is not purification precedent, that does not mean that purification is not possible, but it may be up to you to determine an appropriate purification scheme. Let me know if you have any further questions.
Flow cytometry works for cells or particles and requires a fluorescent tag so the machine can sort according to specific wavelengths. Proteins are not purified by Flow cytometry, as Eric mentioned above.
Proteins are purified by biochemical means using affinity matrices, separation columns, and/or antibodies against the protein. You need to have an assay for your protein function or a good antibody before devising any strategies for purification. Talk to a real Biochemist for advice.