McGraw was born in Madison, New Jersey.[1] He was named after his maternal grandfather, Curtis Whittlesey (1842-1915).[2]: 251 He attended the Lawrenceville School, where he captained the basketball team, and then attended Princeton University, where he was caption of the Princeton Tigers football team.[1]
He later joined the company of his father, James H. McGraw. In 1950, Curtis McGraw became the third successive McGraw to be president of McGraw-Hill Publishing, following his father, James H. McGraw Sr., who served until 1953, and his brother, James H. McGraw Jr., who served from 1935 to 1950.[3]
Curtis became President and Chairman of the Board of McGraw-Hill Publishing on February 24, 1950, when his brother James resigned from those positions.[2]: 488
A resident of Princeton, New Jersey, he died suddenly at the Carlyle Hotel on 10 September 1953.[1] He had been president of the company at the time of his death and was succeeded as president by his brother Donald C. McGraw.[2]: 489