On his own? Probably not. It would have taken someone asking him (begging, really) to serve his country once more. Remember, he died OTL in 1870 from a stroke, and was almost completely indifferent to politics pre-Civil War. An argument could be made that a Confederate victory would have caused less stress on General Lee, thus enabling him to live longer. If Lee is inaugurated in Feb 1868, his term would run til Feb 1874. An interesting possible development is the beginnings of true Confederate political parties. No party would dare risk going against the Hero of the Confederacy, thus whichever party Lee chose to lead would become dominant for at least a generation, the "Party of Lee". And he could well start a dynasty with Fitzhugh Lee becoming President a short time later. But I seriously doubt Lee would willingly choose to become President without major Confederate players asking him to. Now, I could see Lee holding a staff position within the Confederate government, kind of like a precursor to the Joint Chiefs. (I realize that I'm breaking the cardinal rule of A Southern Victory in saying that Robert E. Lee probably wouldn't become President, but whatever.)
But what it really boils down to, is do you want Robert E. Lee to become President? It's your TL. Your history.