The Dallas, which recently delivered humanitarian aid to Georgia's Black Sea port of Batumi, docked on Monday at the Crimean port, where Russia has a naval base, at the invitation of Kiev.
The ship's arrival was met by thousands of anti-NATO protesters chanting "Yankees go home!" and waving banners with the slogan "NATO Stop!" Police cordoned off the area around the ship.
Ukrainian customs officers who boarded the ship and met the commander said they had been prepared to lay on buses for the U.S. crew to give them a tour of the city, but apart from a few officers, no one left the vessel.
Tensions between Russia and the West have been exacerbated by the build up in the Black Sea of U.S. and NATO naval vessels delivering humanitarian aid to Georgia. In an apparent response, Russia sent a group of warships last week, including the Moskva missile cruiser, to Sukhumi, the capital of Abkhazia.
A Russian warship, the Smetlivy patroller, has meanwhile returned to Sevastopol after being involved in peacekeeping operations off the Abkhazian shore, a Russian Black Sea Fleet command source said.
"Smetlivy returned to its base this morning. Everyone on board is safe and sound. A group of ships has remained near the Abkhazian shore to ensure the republic's maritime security," the source said.
Most of Russia's naval group have returned to the Black Sea bases of Novorossiisk and Sevastopol.
After a Russian ship patrolling Abkhazian waters sank a Georgian missile boat during armed conflict last month, Ukraine's Foreign Ministry said Russian warships involved in the operation near Georgia could be prohibited from returning to Sevastopol.
Russia's Black Sea Fleet uses the Sevastopol base under agreements signed in 1997. Ukrainian pro-NATO President Victor Yushchenko announced earlier this year that Ukraine would not extend the lease beyond 2017.
Yushchenko signed a decree last month requiring prior notice of all movements by Russian naval vessels and aircraft from the Sevastopol base in the Crimea. Russia views it as a provocation and is likely to resist any Ukrainian attempts to restrict the deployment of its navy.
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Russia has become very adept in playing the diplomatic game, in which victory depends on choosing the right associate or partner. But there are a growing number of claimants to this role in the new horizontal and interdependent world. Aside Syria and Iran, being still important, the new venues for the application of practical diplomacy may well be Ukraine, the East China Sea and Afghanistan.