Why didn't Arabs discover America?

Because they didn't need to. The Arabs were located around major trade networks. The reason why Columbus sailed around the Atlantic was due to the Ottoman asking European ships for a tariff. European ships had to go through Ottoman waters to have access to the spice and silk coming from the Orient. The Europeans wanted to avoid the taxes, so they sailed around.

Arabs, on the other hand, could easily have access to the Spice road and were near enough the Orient to go there directly. Why go through the trouble of sailing on the Atlantic for 6 months to reach India when you can reach India in less then a month?

Actually, no they were not. They stayed close to the shores and did not do much deep water sailing. Also, by they time Islam took over all "advancements" were demoted to conquest and conversion. Stellar navigation, using Greek Charts still served them well, but as you sail south you loos all the bright stars you use for navigation.

The "Arab" solution was not to sail south.

Not having the Progressive renaissance advantages of Nautical Fleets or the certainty that the Earth is a sphere, the shared European renaissance knowledge/s, communications, or technological advances of engineering and mathematics, astronomy, means and motivation lacked.

the Arabian world dominated by Religious Sects, of the Farthest East.

Because the Native Americans discovered it, actually.