Well Islam didn't spread to the new world because the new world was primarily conquered, colonized and settled by Christian powers — Catholic Spain and Portugal, Protestant England and to a lesser extent Catholic France and the Protestant Dutch Republic.

At the time of the age of discovery, North Africa was comprised of weak and unstable Berber Kingdoms like the Wattasids, the Kingdom of Tlemcen and the Sultanate of Tunis. To the east, in Egypt and the Levant the Mamluk Sultanate was collapsing and falling to the ascendant Ottomans who had just conquered Constantinople a few decades earlier and were still establishing themselves.

At that time, the Muslim world was either incapable or uninterested in the new world. Remember that Christopher Columbus wasn't intending to discover a new continent, he was looking for a shorter passage from Iberia to India because at the time (and indeed, for all of history until the Suez Canal was finished in 1869) to only way to get to India from Europe was to either sail around the Cape of Good Hope at the southernmost point of Africa on a journey that took many months or to go overland through the Middle East which was a very dangerous journey in hostile territory. The Muslim world didn't have that problem since they could just sail from a port in the Red Sea to get to India, which means finding a quicker route wasn't necessary.

Interestingly enough, Moroccan Sultan Ahmad al-Mansur expressed interest in spreading Islam to the Americas.

In a 1601 letter to Queen Elizabeth, al-Mansur stated that he hoped that the defeat of Spain would open up the new world to Moroccan colonization. He believed that this would lead to the rise of Islam in the west, and the proclamation of mahdi on both sides of the Atlantic.

Sultan al-Mansur shared his lofty aspirations with Queen Elizabeth in the hopes of securing English aid in trouncing the Spanish, arguing the following case for an Anglo-Moroccan alliance:

And your high estate shall knowe that, in the inhabiting of [the new world] by us and yow, yow shall have a great benefite: first for that those countries of the East are adjoining to many Kinges Moores and infinite nations of our religion; and further, if your power and command shall be seene there with owre armie, all the Moores will joyne and confederate themselves- by the help of God- with us and yow.

Obviously the English weren't buying it. Where al-Mansur saw England as Morocco's stalwart ally against the Spanish menace, the English more or less saw Morocco as a thorn in Spain's side. England had nothing to gain from sending droves of Muslims to the Americas. Thus, al-Mansur's ambitions for an Islamic New World never saw fruition.

In addition to the excellent answers above, it is important to remember that Islam did spread to the New World through African slaves, just not in the same numbers as Christianity. I am currently moving and don't have my full library, but an excellent source on this subject is Michael Gomez "Exchanging our country marks."