Churkin was speaking at an emergency United Nations Security Council meeting on Ukraine on March 3, which Russia called for.
He quoted from the letter, dated March 1: "I would call on the president of Russia, Mr. Putin, asking him to use the armed forces of the Russian Federation to establish legitimacy, peace, law and order, stability and defending the people of Ukraine."
Russia has poured troops into Ukraine's Crimea region, sparking outrage and threats of sanctions from the United States and European Union.
Western envoys to the UN rejected Russia's justification, with U.S. Ambassador Samantha Power saying, "The Russian mobilization is a response to an imaginary threat."
Ukraine’s UN Ambassador Yuriy Sergeyev said Russia had deployed some 16,000 troops to the region since last week.
Meanwhile, in Crimea, a reported ultimatum issued by the commander of Russia's Black Sea Fleet, Aleksandr Vitko, for Ukrainian military forces stationed in Crimea to surrender or face attack has passed, with no signs of fighting.
News of the ultimatum -- reportedly for 5 a.m. local time -- came from Ukraine's Defense Ministry on March 3. But Russian Defense Ministry spokesman Vladimir Anikin said late on March 3 that no such ultimatum had been issued, and Interfax quoted a Black Sea Fleet spokesman as saying such claims were "utter nonsense."
And in Russia, President Vladimir Putin has ordered all troops that took part in surprise military exercises near Ukraine to return to their bases.
Russian news agencies quoted Kremlin spokesman Dmitry Peskov as saying on March 4 that the drills had been a success. Putin personally observed the exercises on the Kirillovsky training ground in the Leningrad region, together with Defense Minister Sergei Shoigu, on March 3.
Putin ordered the drills across western Russia last week amid high tensions in Ukraine following the ouster of pro-Russian President Viktor Yanukovych. Moscow denied that they were linked to the crisis in Ukraine.
The state of Ukraine's armed forces on the Crimean Peninsula is unclear.
The interim government in Kyiv appointed Denis Berezovsky to be commander of the Navy but Berezovsky announced on March 2 he was supporting the pro-Russian self-declared Crimean government.
Berezovsky was quickly dismissed and as of March 3 naval officers were reportedly backing Kyiv's new selection for commander, Serhiy Haiduk.
Ukraine's acting President Oleksander Turchynov said Russia's military presence was growing in Crimea and urged Moscow to halt what it called aggression and piracy.
Turchynov said the situation was difficult in the south and east of the country, where there are many Russian-speakers, but that the Ukrainian authorities had matters there under control.
He told a news briefing that Russia's Black Sea Fleet had trapped Ukrainian Navy vessels in the bay of Sevastopol, where the Russian fleet has a base
U.S. Prepares Ukraine Support
In Washington, U.S. President Barack Obama said Russia was "on the wrong side of history" on Ukraine and that Russian actions violated international law.
Obama said Russia's decision to deploy troops in the Crimean Peninsula would prove a "costly proposition for Moscow." He urged Putin to allow international monitors to mediate a deal that would be acceptable to Ukraine's people.
The Pentagon said it had "put on hold" all military-to-military engagements between the United States and Russia, including exercises, bilateral meetings, port visits, and planning conferences.
The chairman of the U.S. Senate Foreign Relations Committee, Robert Menendez, said the panel was preparing legislation to provide at least $1 billion in loan guarantees to provide structural support to Ukraine's economy.
Menendez said the committee was also consulting with the Obama administration on possible sanctions against individual Russians, and Ukrainians cooperating with them.
The sanction would range from visa bans and asset freezes to suspending military cooperation and sales, as well as economic sanctions.
Russia and Western nations have exchanged sharp criticisms over the situation in Ukraine.
The top U.S. diplomat in Europe says that "advance teams" from the Organization for Security and Cooperation in Europe (OSCE) will start deploying in Ukraine late on March 3, but Russian objections mean the pan-European rights body has yet to agree on a full-scale mission.
U.S. Assistant Secretary of State for European and Eurasian Affairs Victoria Nuland, talking in Vienna on the sidelines of a special meeting of the OSCE, called on Russia to make "the right choice."
Secretary of State John Kerry is heading to Kyiv to lend support to Ukraine's interim leaders.
WATCH: Thousands protest Russian actions in Ukraine's Sumy.
EU foreign-policy chief Catherine Ashton says European Union member states have agreed to consider targeted sanctions against Russia if Moscow does not "deescalate" the threat of military action against Ukraine.
Ashton, speaking after an emergency meeting of EU foreign ministers on March 3 in Brussels, said the EU strongly condemned what she called the "clear violation" of Ukraine's sovereignty and territorial integrity by Russia.
She called on Russia to withdraw its troops back to bases, and urged Moscow to agree to Ukraine's request for consultations. Ashton said she would hold talks with Russian Foreign Minister Sergei Lavrov on March 4 in Madrid, before traveling to Kyiv on March 5.
Speaking at the UN Human Rights Council in Geneva, Lavrov criticized threats of "sanctions and boycotts." "I call upon them to show responsibility and to set aside pure political calculations and put the interest of the Ukrainian people above all." Lavrov said.
NATO allies will hold emergency talks on the crisis in Ukraine on March 4 for the second time in three days.
WATCH: Troops, Believed To Be Russian, Surround Ukrainian Base In Crimea