In theory, and under the provisions of the Constitution of the United States, there are two ways that U.S. participation in an armed conflict can be brought to an end. Article I, Section 8 of the Constitution vests with the Legislative Branch of government the power to declare war. This same section also states that
"The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States." [emphasis added]
Article II, Section 2, assigns responsibility and authority for commanding troops committed to war to the Chief Executive, in effect, the president of the United States. Specifically, that provision declares:
"The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States."
This division of responsibilities and powers within the Constitution was deliberate, part of the document's authors' determination to prevent the consolidation in any one branch of government of undue power to entangle the United States in foreign commitments, especially wars. The student's question, though, raises an important constitutional point that has not been resolved. Having spent many years working for members of the Armed Services Committees of the House of Representatives and the Senate, during which this very question was repeatedly debated, I can confidently state that the provisions of the Constitution intended to affect the manner in which the United States extricates itself from foreign conflicts does not, in fact, work as intended.
The reason for this cynicism regarding decisions on terminating U.S. involvement in overseas military conflicts stems from the extreme reluctance of most members of Congress to exercise its most important authority with regard to military affairs: the budget. The reference above to the Constitution's provision granting to the Legislature the power over federal spending--in effect, "the purse-strings"--and its responsibility to "provide for the common defense" is that branch of government's most important responsibility. And Congress does wield that authority when it comes to shaping how the armed forces are structured and equipped. Congress has often been reluctant to use its powers is in the area of war termination. In other words, most senators and congressmen are extremely wary of forcing presidents to withdraw troops from overseas contingencies by cutting off the supply of money needed to support those troops in their mission.
This educator was involved in staffing members of the United States Senate when that institution debated whether to cut off funding for U.S. troops engaged in conflict in the former Yugoslavia, as well as for other contingencies. In each instance, debate over the merits of exercising that authority ended with no decision made by Congress to exercise its constitutional prerogative. Members of Congress simply do not want to be seen as denying support to troops deployed in a war zone. Which brings us to the practical answer to the question of who decides to end American military participation in a war. The obvious answer, then, is the president. Congress can declare war, but it cannot order troops into combat. Only the president of the United States has that authority. And only the president of the United States can decide when to bring those soldiers home. Congress can, as discussed, force the issue by cutting off funding, but it will not take that step in the overwhelming number of contingencies. The war, or, at least, U.S. participation in the war, effectively ends when the president says it does.
This discussion, clearly, is specific to U.S. involvement in a war or other military contingency. U.S. withdrawal from a conflict does not necessarily mean the end of the war, only the end of a direct U.S. role as a combatant. The war itself might continue on for many more years.