Was Columbus trying to prove the Earth was round?

No. That the Earth is a sphere had been established knowledge among educated Europeans for nearly two thousand years before Columbus sailed, and nobody he petitioned believed otherwise.

The ancient Greeks worked it out. Eratosthenes calculated the Earth's circumference around 240 BC using the angle of shadows at two different latitudes, and came remarkably close. Aristotle had already argued for a spherical Earth from the curved shadow it casts on the Moon. That knowledge was preserved through the medieval period, not lost — medieval scholars represented the Earth as a sphere routinely, and the orb held by monarchs in coronation imagery depicts exactly that.

The real dispute with Columbus was about size, and his opponents were right.

Columbus used a set of figures that made the Earth far smaller than it is and Asia far wider than it is. By his arithmetic, Japan lay roughly 2,400 nautical miles west of the Canary Islands. The actual distance is over 10,000. The Spanish and Portuguese advisers who rejected his proposal did so because they knew the ocean was too wide to cross with the provisions of the day — and on that they were correct. He survived because an unmapped continent happened to be in the way.

The flat-Earth story was largely popularised in the nineteenth century, notably by Washington Irving's 1828 biography, which invented dramatic scenes of Columbus facing down flat-Earth clergy. It suited a period keen to portray the medieval church as the enemy of science.

Columbus also died insisting he had reached Asia, never accepting that he had found a separate landmass.