Shaw was first taught in a local national school, and later by James Rowan, Presbyterian minister of Kircubbin.
In 1861 he entered Queen's College, Belfast, gaining the highest entrance scholarship in classics, the first of many honours.
After studying theology in the General Assembly's College, Belfast, and at the University of Edinburgh, he was licensed to preach in 1869 by the Presbytery of Ards, and was appointed in the same year by the General Assembly Professor of Metaphysics and Ethics in Magee College, Derry.
In 1902, he joined the council of trustees of the National Library of Ireland, and in 1908 was chairman of a viceregal commission of inquiry into the mysterious disappearance of the Irish Crown Jewels from Dublin Castle.
When the Queen's University of Belfast was founded by royal charter in 1908, he was appointed by the Crown chairman of the commission charged with the framing of the statutes, and he discharged the duties of this office with marked ability.
A singularly clear thinker and writer, and a high-principled administrator, Shaw died in Dublin on 27 April 1910, and was buried in the Mount Jerome Cemetery there.
Shaw married in 1870 Mary Elizabeth (d. 1908), daughter of William Maxwell of Ballyherly, Portaferry, County Down, by whom he had one daughter, Margaret (who married Robert H. Woods, president of the Royal College of Surgeons in Ireland, 1910–11 and MP for Dublin University, 1918–22), and two sons.