1. The Micronesians were using coconut sextants circa 2000 BC. - By the time Colombus set sail, it was well known that 1 minute of arc subtended 1 nautical mile on the surface of the earth.
2. Columbus believed the earth was round. 360 degrees in a circle and at 1 nm per minute of arc if the earth was indeed round, its circumference was approximately 21,600 nautical miles. - Simple sums.
3. He knew where he was starting from. He knew how far it was (on an easterly route) to 'the Indies'. Had he subtracted that distance from 21,600 it would have given him a rough idea of how far he would have to sail to reach the same point on a westerly heading.
4. He knew the average speed of his ships and he should have known how many nautical miles he could expect to cover in a day.
5. He knew how many days he had been at sea when he arrived at the 'West Indies'.
Brave man? - Certainly.
Intrepid explorer? - Certainly.
Navigator? - Hardly. He made the biggest DR error in history.
As for 'discovering' America, well it already had people when he got there, so he could hardly claim to have discovered it.
If you sail west from Europe (and if you don't sink) then eventually your ship will stop sailing - You will run out of sea. He didn't 'discover' America, he ran into it.
He named the land he 'discovered' after his sponsor. Fortunately for the inhabitants of that continent, Columbus appears to have been an informal sort of bloke and he named it after that sponsor's first name.
Otherwise, they would be Vespuccians.
The American Federal holiday named after him is 'Get lost' day.