No, he wasn't.
In fact, the idea that anyone in his time believed the earth was flat is a complete fiction, invented in 1828. There was a very minor debate in Columbus' time, but that was not about whether it was round, but how big it was. And in that debate Columbus happened to be dead wrong. Luckily for him, when he sailed west using his incorrect calculations, he just happened to hit the Caribbean before he ran out of supplies.
A Medieval depiction of the earth as a sphere
The fact is, the Church did not teach that the earth was flat at any time in the Middle Ages. Medieval scholars were well aware of the scientific arguments of the Greeks that proved the earth was round, thanks to their knowledge of Plato's Timaeus, which shaped their views of the cosmos, and later their use of Ptolemy and Arabic astronomers. And they could use scientific instruments, like the astrolabe, to accurately measure its circumference. The standard text on astronomy used in medieval universities from the later Thirteenth Century was John Sacrobosco's De sphaera mundi - "the Sphere of the World". The fact that the earth is a sphere was so well known, widely accepted, and unremarkable that when Thomas Aquinas wanted to choose an objective fact that is not able to be disputed early in his Summa Theologica he chose the fact that the earth is round as his example.
And it was not only the learned who knew the shape of the earth - all evidence indicates that this was commonly understood by everyone. A symbol of the earthly power of kings, used in their coronations, was the orb: a golden sphere held in the king's left hand to represent the earth. That symbolism would not make sense if it was not understood that the earth was round. A collection of German sermons for parish priests from the Thirteenth Century also mentions, in passing, that the earth was "round like a apple" with the expectation that the peasants hearing the sermon already understood what this meant. And the popular Fourteenth Century English book of travelers' tales, The Tales of Sir John Mandeville, tells of a man who traveled so far east that he returned to his homeland from the west, while not explaining to its audience how this works.
The popular idea that Christopher Columbus discovered the earth was round and that his voyage was opposed by the Church is a modern myth created in 1828. The novelist Washington Irving was commissioned to write a biography of Columbus, with the brief that he depict Columbus as a radical thinker who turned his back on the superstitions of the old world. Unfortunately, Irving found that Columbus was actually wildly wrong about the size of the earth and discovered America by pure chance. Since this did not make a very heroic story, he invented the idea that the Medieval Church taught the earth was flat and created this persistent myth when his book became a best-seller.
The fact that many people (including some of the poorly supported, assertion-laden answers to this question) continue to believe in the myth that people in the Middle Ages were taught the earth was flat shows how pernicious this kind of tendentious myth can be.More questions on Christopher Columbus (explorer):