Personally, I do not think so. For example (I'll expand on this I realize the mayflower didn't discover the New World) the time the Pilgrims landed (1620) the 80 years war had begun, along with the Thirty Years war, and the internal religious conflicts between protestants and catholics in France had subsided for the time. Documents such as the peace of augsburg had been signed by that time. I'm using the Pilgrims as a sort of Benchmark because they were some of the first religiously persecuted Europeans to successfully colonize an area of the New World.

Meanwhile, other than the Pilgrims most other attempts to land colonizing attempts by religious minorities I at least am aware of were conducted by Huguenots (french protestants) in the Southern United States. The Spanish tried to stamp these out whenever possible (burning of Fort Caroline). Largely the colonial game at that time was dominated by the Spanish and Portuguese, both staunchly, aggressively catholic, and they had essentially divvied up the entirety of the New world with the assistance of the Pope (Line of Demarcation) and were sometimes forcibly converting natives in their new dominions.

The fleeing of the persecuted (Quakers, Pilgrim types) and establishment of religious refuges like Maryland for Catholics comes somewhat after some major religious conflicts in Europe that had created some freedoms.

I feel like i'm missing something vital here though... but overall I would say the birth of religious freedom in the West (at least for christians) began in Germany and the Lowlands where the conflict was perhaps the greatest between Protestants and Catholics.