Believing he was traveling to Asia, Columbus particularly wanted to convince the Grand Khan of China, who had already expressed interest in Christianity, to convert. He thought that the Grand Khan could help with the crusade to take Jerusalem by marching from the east, while the Europeans marched from the west — an interesting idea.
Now that makes it a bit more complicated, as the "Grand Khan" is probably a reference to Dayan Khan who was Mongolian ... but definitely not Indian.
Definitely would have been a major embarrassment had Columbus managed to get to Beijing and started asking for the Mongolian Khan.
But newly converted Christian Mongols invading Islamic controlled Jerusalem is one of those too cool for movies idea.
MONGOLIAN ELEPHANTS WIN.