Africa or westward as Columbus intended. When Columbus's expedition was successful despite his not reaching Asia, it was one of the great turning-points of history. At first Columbus's and the following expeditions did not have the intention of converting the populations of the "unknown world" to Christianity. The Europeans were actually more interested in exploiting them for slave labour. But once the indigenous American peoples were discovered, the idea of conversion, to "save their souls," became a convenient rationale for the exploitation of these people and the resources of their lands. And it's nevertheless true that many devout Europeans sincerely believed that conversion of non-western peoples to Christianity was a necessity precisely because according to Christianity it was the only route to eternal salvation.

In their dealings with the Native peoples, the Spanish were cruel to a genocidal degree. Throughout the Americas, more indigenous people were killed by diseases the Europeans carried, such as smallpox, than were killed by arms. Of the colonial powers, the Spanish intermarried with the Native Americans to a degree the northern European nationalities did not. Today, a majority of Latin Americans are of mixed Amerindian and European descent. This was an adjunct to the forced conversion of the Native peoples, since through intermarriage they were made to adopt Christianity.

The French perhaps had the least cruel or controlling campaign in America, wishing chiefly to trade with the American Indians rather than to subjugate them. As with the Spanish, however, the French sent their clergymen and missionaries to convert them. Still, the main purpose oof cultural contact was enrichment through trade and, as it was more generally in Spanish America, theft.

Of the three major European groups, the English were those who least wished to intermix with the Native Americans. They also wished, more than the French, to establish large colonies of their own people who for many reasons, often religious, were marginalized and persecuted by the English ruling class. Religious freedom for Puritans, Quakers, and other dissenters was thus the rationale to colonize North America. Colonization was not for the benefit of the indigenous Americans but for the English themselves. The English had less influence in converting the Native Peoples than did either the Spanish or the French. The goal of the English and later the Americans was, in a process that continued for over 250 years, chiefly to push the Native Americans into the interior of the continent and eventually grant their remaining populations isolated tracts of land, "reservations" or "reserves."

But all three of the European nations, even the English to an extent, had religious conversion at least as an indirect goal or as a rationalized form of their motives for self-aggrandizement and the expansion of their own wealth and power.

I would argue that European interest in the New World had much less to do with creating a religious utopia and more to do with interests in exploration and material exploitation. 

For the French and the Spanish, Christianity was merely the tool that was used to justify conquest. Before race was invented and used as a tool for conquest, Europeans created the division of Christian vs. heathen, or savage. Native Americans, because they had not been baptized as Christians, were the savages. Their absence of Christian faith and Christian manners provided, in the minds of their conquerors, the justification for their exploitation, enslavement, and massacre.

For the English, things were a bit different. By the mid-16th century, the Anglican Church had been formed by Henry VIII. His daughter, Elizabeth I, would rule as a Protestant queen and spent much of her reign fending off infiltration by Catholics, including her cousin, Mary, Queen of Scots. 

The Protestant Reformation resulted in the creation of numerous new Christian sects, though Catholicism remained firmly implanted in France and certainly in Spain. In England, the Pilgrim and Puritan sects were established. They inevitably faced persecution. Their response was to leave England. 

William Bradford was a Separatist leader, and later a signatory of the Mayflower Compact. He had grown up in Yorkshire, England. For a time, he lived in Leiden, Holland with his fellow Pilgrims, believing that, perhaps they could find refuge there. 

His vision, however, expanded into not only into seeking refuge for his sect, but also in creating a religious paradise in the New World. When he and his fellow Pilgrims established Plymouth Colony, they set up an agrarian community, in which they all shared the fruits of their labor. As the governor of Plymouth Colony, Bradford set up a government that separated religious concerns from secular ones. This premise stands in sharp contrast to the kind of government that was established by the Puritans in Massachusetts Bay Colony. The Puritans began arriving in the New World shortly after the Pilgrims. The Pilgrim community also tolerated dissent, whereas the Puritans did not. 

I'm summarizing a lot of background here to help you understand that the French and the Spanish had a very different purpose compared to the early English settlers. The French and the Spanish entered the New World looking for material gain. It is important to note that the French established a very lucrative fur trade in Quebec and parts of Northern New England. They merely used Christianity as an excuse to justify their exploitation of territory and their massacre of natives.

The essayist Michel de Montaigne wrote a short essay on this subject in 1580. The English title is "On Cannibals." I would recommend reading this to give you some additional context. Montaigne uses the treatment of the natives by Europeans as a lens through which he views his society. He writes about the way in which Christianity has been used as justification for murder and exploitation by the French.