Final answer:
Imperialists and anti-imperialists disagreed on the issue of American overseas expansion, with imperialists justifying this with ideas of national destiny, while anti-imperialists opposed it on democratic, economic, and moral grounds.
Step-by-step explanation:
One idea over which imperialists and anti-imperialists disagreed was the concept of American overseas expansion and the establishment of an empire. Imperialists believed in the merits of acquiring territories for economic, strategic, and ideological reasons. They argued that imperialism was justified by a sense of national superiority and destiny, often referred to as American exceptionalism. Meanwhile, anti-imperialists, such as the members of the Anti-Imperialist League, opposed these expansions on various grounds, including the principle that it contradicted the nation's democratic and freedom-based ideals, concerns about competition from foreign workers, and the moral and humanitarian costs of empire building. The disagreement encapsulated political, social, and economic perspectives on U.S. foreign policy, highlighting sharply contrasting visions for the nation's role in global affairs.