Final answer:
John C. Calhoun, concerned about the resurfacing of national debates on slavery, opposed the annexation of Mexican territories because it would disrupt the balance on the slavery issue.
Step-by-step explanation:
John C. Calhoun opposed the annexation of Mexican territories south of the Rio Grande because he feared it would reignite debates about slavery in the newly acquired territories, leading to intensified sectional divides.
This opposition arose during the period surrounding the Mexican-American War, where the potential expansion of the United States into these territories threatened to disrupt the tenuous balance between free and slave states. Calhoun, aware of the contentious nature of slavery within the national political discourse, particularly after the gag rule in Congress that tried to minimize debates about it, understood that the annexation and the subsequent discussions over the status of slavery in these regions would raise complex and divisive issues.
This proved true as the Treaty of Guadalupe Hidalgo and the debates surrounding the Wilmot Proviso significantly heightened sectional tensions.