Nobody can ever unite the country. They can only cobble together a decently sized chunk of it for a certain period of time.
I think a true independent could unite the country. Maybe if James Mattis ran for President he could unite it, but I think he has no interest in being president.
I don't think it is possible. A good 1/3 of Americans are solid right wing and anybody who isn't unabashedly right wing they will hate, but if you choose someone unabashedly right wing you alienate over half the country. Likewise there is a growing chunk of progressives who want the president to be a true progressive not centrist, but the US is still a centre-right country so if one tries to copy progressive parties in Canada and Europe you will lose support from the majority.