In a statement posted on Telegram late on Monday, Bzhania's press office said Abkhazia's parliament would consider his resignation on Tuesday.
It said Bzhania's deputy would become Abkhazia's acting leader, who would then dismiss the current prime minister.
The statement warned that Bzhania would withdraw his resignation offer if protesters refused to leave the government buildings.
Bzhania describes himself as Abkhazia's president - Georgia and most of the world do not recognise his authority.
Abkhazia's opposition has not commented on Bzhania's resignation offer.
Located along the Black Sea and Caucasus mountains, Abkhazia is known for its natural beauty and coastline, and was one of the favoured holiday destinations for the Soviet elite until the collapse of the USSR in 1991.
Locals feared that the proposed bill would trigger an influx of Russian investment, pricing them out of the property market and further strengthening Moscow's grip on the region.
Backed by Russia, Abkhazia fought a war of secession with Georgia in 1992-93, before unilaterally declaring independence.
Abkhazia's economy depends overwhelmingly on tourism from Russia.
In 2009, Moscow signed a five-year agreement with Abkhazia to take formal control of its frontiers with the rest of Georgia, while in 2014 Moscow and Sukhumi signed a "strategic partnership" agreement.