Often folk assume that the first Christians in America were either with Christopher Columbus, or (with Orthodox) the Russians in Alaska. However, the tale of Christianity in America really begins with the Gaels and Vikings.
Greenland is part of North America, though not attached to the mainland. When Leif Ericsson (a Greenlander, so an American) made his first journey, it was to Norway where he received the faith from St. Olaf, the King of Norway. His mother was also a Christian (many of the Icelanders who settled Greenland were Christians, as the first 'Norse' in Greenland where Hebridean Norse that held to Irish Christianity - especially the paruchia of St. Columba.) St. Olaf placed the Vestmenn (West-men, ie Americans) under the Archbishop of Hamburg-Bremen (today in Germany between Holland and Denmark - in those days, part of Greater Frisia.) They took priests with them back to Greenland, and also on to Vinland. This all, of course, is before the Great Schism. Hamburg-Bremen at the time was part of the larger Northern Church with its center at Canterbury (during the reign of Cnut Dane, King of England, Denmark, Norway and part of the Swedes.)
An early Christian on the mainland was Thorvald, who was killed in an encounter with the Skraelings (possibly the Beothuk or Micmac Algonquins.) His burial place was named 'Crossness' by the Greenlander West-men, and thus is the place of first Christian burial in mainland Americas.
Even earlier is the tale of Ari Marson, who in 983 was captured by natives when blown off course to 'Vinland'. It was said that the later Viking explorers came across Ari again - in Hvitramannaland (White-robed men's land) also called Great Ireland "...behind and interior to Vinland." Ari had been baptized there by Irish monks, and made king over the natives (mixed Indian and Irish.) An Icelandic witness to St. Brendan the Navigator and other Irish indeed coming to the New World?
Eventually, a bishopric for the New World was established (during the period when the schism was developing, but before it the 'Sack of Constantinople' and its finality of excommunication) - this bishopric continued down to the early decades of the 16th century - the Greenland colony (with visits to Vinland) overlapping Columbus' voyages. The final end of the Greenland colony was due some believe to a 'mini-Ice Age', though local legends and contemporary Scandinavian records blame the depredations of English and Basque pirates, and finally the end of the few survivors at the hands of the Inuit (who were also recent in that area.) Between the end of the Christian Scandinavian culture of North America (c. 1520) and the founding of the English Christian colonies (1582) was roughly two generations of possibly Scandinavian influenced exploration and visits for potash, lumber, furs and fish by English enterprise. The first British colony was actually at Vinland (though it failed) though it overlapped the foundation of the Scots colony of Annapolis Royal about two decades later - which survived til today as the first permanent English speaking colony (Fort Popham, Maine and Jamestown, VA were founded only two years later.) Thus there is a thin thread of continuity before the Anglican settlement of Virginia (which was less 'Protestant' ie Reformed/Calvinist/Congregationalist/Puritan than the New England colonies a decade and a half later.)
The rite of worship of these Scandinavians (Orthodox Catholic Christians) of America was that of Trondheim, being a local adaptation of the use of Sarum ... the English who first colonized America being only one to two generations removed from full Sarum (and for recusants, not removed at all) and at least following the new English BCP's which were also adaptations of the Sarum. So - a continuity of worship with the first Christians on the continent from the days before the 'Great Schism'
(Another interesting note: the old usages of the Celtic/Anglo-Saxon church were preserved at Sherbourne Abbey, and became part of the Sarum usage at Salisbury in Norman times - the temporal lord over Sherbourne was none other than Sir Walter Raleigh, who may be considered in more than one way the 'Father of the American colonies'. America, of course, was named so before Amerigo Vespucci made his voyage or maps - the local tradition being that it was named so for the Bristol harbor-master, Richard Americ by Cabot on his voyage of 1497 - just 5 years after Columbus first voyage, and while the Greenland settlement still existed. Vespucci, whose real name was Alberrico, not Amerigo til a name change in the early 16th c., did not make his first voyage until 2 years after Cabot and the naming of America.)