Sergei Bagapsh said he expected South Ossetia, another secessionist republic in Georgia, to send a similar request in the coming days.
In Abkhazia's capital, Sukhumi, more than 47,000 people gathered Thursday on the central square to show support for the appeal.
"It has become clear that Abkhazia will not live in the same state with Georgia," parliament speaker Nugzar Ashuba told the rally.
The appeal urges Russia to recognize Abkhazia "as a sovereign and independent state, and sign a treaty of friendship, cooperation and mutual assistance."
"Considering that the threat of Georgian aggression is still very serious... we believe that it is necessary to include into this treaty a provision on the further presence of Russian troops in Abkhaz territory," the appeal said.
On Wednesday the proclamation was backed by all 26 MPs attending an extraordinary session of Abkhazia's parliament.
The MPs also adopted an appeal to the parliaments and governments of other countries.
The recognition request was prepared shortly after Georgia launched an offensive on South Ossetia on August 8.
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Ukraine has not preserved its 1991 borders. The signing of the Geneva memorandum on April 17 reaffirmed the willingness of Russia, the United States and EU countries to reach a compromise. While the sides continue to trade tough talk and symbolic sanctions, the Kremlin and the White House are also holding a parallel dialogue on the coordinated geopolitical revision of Eastern Europe.