Monroe University is a private for-profit college in New York City. It was founded in 1933 and has campuses in the Bronx, New Rochelle and Saint Lucia, with degree programs also available through Monroe Online. The college is named after James Monroe, the fifth president of the United States. It is accredited by the Middle States Commission on Higher E…Monroe University is a private for-profit college in New York City. It was founded in 1933 and has campuses in the Bronx, New Rochelle and Saint Lucia, with degree programs also available through Monroe Online. The college is named after James Monroe, the fifth president of the United States. It is accredited by the Middle States Commission on Higher Education.