Let's define our terms a little better.
Public schools are those that are, ultimately, run by "the government" (in the US, by a state government). They get money from the state to operate (they also charge fees, and usually out-of-state students are charged extra).
Private schools are those run by some organization other than the government. They usually charge higher fees, but don't charge out-of-state students extra.
The University of Phoenix is a private school with an added twist: it's actually run for profit (all public schools, and many private schools, just try to break even).
The University of Phoenix is a for-profit institution of education. It would not be considered a public school, because it is not funded by any particular state. It is based in Phoenix, Arizona, but it has over 200 campuses worldwide. It is a wholly owned subsidiary of Apollo Group Inc.
public
public
The school is a private university.
Harvard is a private school.
If you mean the University of Phoenix, it is an online school. If you mean any other school with Phoenix in its name, I would assume it is in Phoenix, Arizona.
Strayer University is a private, for-profit educational institution.
Yes, Grambling State University is a private school
Probably not. A strong argument could be made for saying that either Harvard or Yale was the top private university in the US.
Johnson and Wales University is a private, nonprofit University.
The University of Dallas is a private, non-profit, Catholic university.
A private university in Maryland