Like a lot of end-of-the-world believers, he based his prediction on the Book of Revelations.

From Wikipedia:
“To understand the reasoning behind the 2060 prediction, an understanding of Newton’s theological beliefs should be taken into account, particularly his antitrinitarian beliefs and those negative views he held about the Papacy. Both of these lay essential to his calculations, which are themselves based upon specific chronological dates which he believed had already transpired and were prophesied within the Book of Revelation and the Book of Daniel within the Christian Bible.

Despite the dramatic nature of a prediction of the end of the world, Newton may not have been referring to the 2060 date as a destructive act resulting in the annihilation of the earth and its inhabitants, but rather one in which he believed the world was to be replaced with a new one based upon a transition to an era of divinely inspired peace. In Christian and Islamic theology this concept is often referred to as The Second Coming of Jesus Christ and the establishment of The Kingdom of God on Earth.”

So basically he took writings from many many many years ago that were based on dreams, added a dash of his own personal politics and self-importance and came up with a different date from the other hundreds (thousands) of predictions.