An American soldier's alleged massacre of 16 Afghan civilians has inflamed anger against U.S. troops at a time when the American public -- increasingly including Republicans -- is tiring of America's decade-long presence in Afghanistan.

An ABC News/Washington Post poll conducted before the killings underscored deepening cynicism about the Afghanistan war, with 60 percent of Americans saying the war has not been worth fighting: 44 percent of those surveyed felt strongly that the war has been a mistake, versus a mere 14 percent who felt strongly it was necessary.

A backlash against the civilian slaughter has endangered the Obama administration's goal of cultivating an alliance with Afghan security forces, but Americans are skeptical that such cooperation can exist: 55 percent believe that Afghans do not support the American mission there.

While self-identified Democrats and independents dissaproved of the war by a wide margin, Republicans were evenly divided. That reflects a fissure running through the field of Republican presidential candidates: Mitt Romney and Rick Santorum have accused Obama of prematurely pulling troops from Afghanistan for political reasons, while Newt Gingrich acknowledged the war may have been a folly. (Ron Paul, a staunch libertarian, has consistently opposed intervening in foreign countries.)

I think it's very likely that we have lost, tragically lost, the lives and suffered injuries to a considerable number of young Americans on a mission that we're going to discover is not doable, Gingrich said on Fox News Sunday.

Look at the things that are going on around the region and then ask yourself, Gingrich continued, is this in fact a harder, deeper problem that is not going to be susceptible to military force --at least not military force on the scale we're prepared to do?

After initially committing an additional 30,000 troops to Afghanistan in an effort to stabilize the country and beat back a resurgent Taliban, Obama announced in June plans to accelerate the withdrawal of U.S. troops from Afghanistan.