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The 34-year-old's decision means PSG have an extra space in their squad for a non-EU player.
Silva has played 274 games since joining the reigning Ligue 1 champions in 2012 and has 77 caps for Brazil.
"Today me, my wife @Belle_Silva, and my children Isago and Iago have become French citizens. It's a very happy day for us because we have been in Paris for six-and-a-half years and we really feel at home," he said.
"Thanks to France, a country which has welcomed us so much where we have experienced some unforgettable moments. Vive la France," he added.