Using Military Force Through the UN
In the event that it is necessary for the US to use military force, whenever possible, a strong majority prefers to act through the UN rather than unilaterally. A plurality even prefers acting through the UN over acting through NATO.
Americans have consistently shown a strong preference for using military force under the auspices of the United Nations rather than acting multilaterally. As discussed in the Terrorism report (see Terrorism: Support for Multilateral Approaches) an overwhelming majority felt it was important for the US to get UN Security Council approval for taking military action in response to the September 11th attacks. When asked whether the US should proceed with military action if it failed to get UN approval, the public was divided--though it could be argued, based on the UN Charter, that the US would have the right to act unilaterally out of self-defense.
Polls that have asked about the broader principle of using military force through the UN have found overwhelming support. In an April 1995 PIPA poll (even though this was a time when the UN operation in Bosnia was not going well), 89% agreed that:
When there is a problem in the world that requires the use of military force, it is generally best for the US to address the problem together with other nations working through the UN, rather than going it alone.
This attitude was sustained even in the face of a strong challenge that the US would be more successful acting on its own. Only 29% agreed with the argument: "When there is a problem in the world that requires the use of military force, it is better for the US to act on its own rather than working through the UN because the US can move more quickly and probably more successfully." Sixty-six percent rejected it.
PIPA asked respondents in June 1996: "As a general rule, when it is necessary to use military force to deal with trouble spots in the world, do you feel more comfortable having the US contribute to a UN military action or for the US to take military action by itself?" Sixty-nine percent preferred the US to contribute to a UN action, while only 24% preferred the US to act alone.
ATIF asked in June 1995, "When faced with problems involving aggression, who do you think should be 'policeman to the world,' the US or the UN?" Only 19% said the US, while 76% said the UN. ATIF also asked, "When faced with future problems involving aggression, who should take the lead, the US or the UN?" In June 1995, 69% said the UN, down from 85% in March 1991, shortly after the Gulf War. 
The majority is also strikingly responsive to the idea that the US should restrain itself from using force unilaterally even when it is in the US national interest, preferring the US to limit its use of force to multilateral efforts through the UN. The Los Angeles Times asked in December 1993, "Should the US try to use force only in concert with the United Nations, or should the US use force in our own national interest regardless of the United Nations?" Fifty-nine percent said the US should act only with the UN in using force, while just 31% said the US should use force to pursue national interests. In a December 1992 Newsweek poll, an extraordinary 87% agreed that "the US should commit its troops only as part of a United Nations operation."  While this finding does demonstrate how strongly Americans prefer multilateral force, it is unlikely that such a large number really meant that the United States should never use force unilaterally. As mentioned above, when asked about taking military action in response to the September 11th attacks without UN Security Council approval, the public was divided.
This preference of the public for acting through the UN rather than unilaterally leads to support for UN peacekeeping. UN peacekeeping is seen as a means of burden sharing. In an April 1995 PIPA poll, an overwhelming 86% agreed that: "The only way for the US to not always be the world policeman is to allow the UN the means to perform some policing functions. UN peacekeeping is a way we can share the burden with other countries." Seventy-six percent agreed in PIPA’s February 1994 poll. 
Consistent with this thinking, when questions are posed about the US sending troops to trouble spots, Americans are very sensitive to whether the wording of the question implies a unilateral or a UN operation. Questions that simply ask about sending US troops to a destination nearly always elicit opposition. However, questions that ask about contributing US troops to a UN operation usually elicit a majority or at least a divided response.
Mild Preference for UN over NATO
What evidence there is suggests that, if anything, Americans lean toward preferring the UN over NATO as a vehicle for using US military force. In July 2000, COPA asked whether "As a general rule, when it is necessary for the US to use military force, do you think it is best for the US to act as part of a United Nations operation, act as part of a NATO operation, or act on its own?" A 49% plurality preferred the US to act as part of a UN operation; 26% preferred NATO; and only 17% preferred the US to act on its own. In PIPA’s April 1994 poll, respondents were asked about their preferences for a peacekeeping force to enforce a peace agreement in Bosnia. Asked "Would you prefer that this force be under UN command, NATO command, or would you say it does not matter much to you?" a plurality of 39% chose UN command, 25% chose NATO, and 29% were indifferent. 
In the Wirthlin Group’s December 1995 poll—taken just before the deployment of the NATO-run Bosnia peacekeeping force—respondents were told that "the warring parties have just agreed on a peace settlement" and were then asked which of three following options they would prefer for Bosnia: "A military force, including some Americans, run by NATO to enforce that peace settlement; a peacekeeping force, including some Americans, under the United Nations that would monitor that peace settlement; or no organized international presence..." Fifty percent thought a force under the UN would be the best option. Only 17% favored a NATO-run force, while 25% opted for "no organized international presence." 
A possible problem with this finding, though, is that each of these command arrangements was also associated with a different form of engagement—the UN command was associated with monitoring, while the NATO command was associated with enforcement. However, it is unlikely that this factor determined the outcome, because other data suggest that more Americans would prefer enforcement over simple monitoring. In April 1995—seven months before the Dayton accord—PIPA asked respondents to imagine that the warring parties in Bosnia came to a peace agreement and the UN agreed to police it. In such an event, 62% thought that "if one side violates the agreement," the peacekeepers "should be able to use military force to force [the violating side] into compliance," while just 34% thought that "UN peacekeepers should simply monitor the borders between the parties." 
The same Wirthlin group poll also asked respondents what kind of response they would like to see to conflicts in the former Yugoslavia, Rwanda, and Haiti. Given three options, only 14% chose the option closest to working through NATO—having "a few big countries like the United States and those in Europe deal with the situation on their own." A plurality of 48% opted for "the UN tak[ing] the lead," while 30% said that "the outside world [should] just stay out." 
Putting all these findings together, it appears that the public’s comparative comfort with the UN imprimatur is a significant factor that can outweigh popular frustration with the lack of assertiveness often found in UN peacekeeping.