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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The project of a Eurasian Union can be considered as a response to the consequences of neo-liberal globalisation, which led to economic and moral decline in the countries forming the Commonwealth of Independent States. It is part of a more general movement in world politics towards regionalisation.