How did religious disputes mold both European and American history?
This answer is going to be general in design, leaving you room to fill in the details of specific religious movements and ideas on your own.
Religion has always been at the root of any society's formation because it has such power in forming the thoughts and practices of large groups. The pursuit of religion is among the main causes for people to leave homes, accept certain types of jobs, pass specific rules and legislation. It brings about a sense of community and unity that is unsurpassed by any other type of group activity.
Whenever a religious movement splits it is an "earth moving" experience. Whether it is the split between the Roman Catholic church and the Church of England which affected life in all of Europe or the Pilgrims and Puritans fleeing Europe for America because of the pressure from the Catholic Church, these movements have resulted in the production of the identity of the inhabitants of Europe and of America as members of "Western Culture."