He reacted after the party confirmed he was the only candidate to succeed Humza Yousaf, who announced one week ago he was stepping down as SNP leader and Scottish First Minister.
Mr Swinney, 60, is now expected to become Scotland's seventh first minister but must first win the backing of MSPs in a vote at Holyrood.
That ballot could take place on Wednesday, with Mr Swinney likely to be officially in place in the top job in Scottish politics the next day.
He posted: "I will give all that I have to serve my party and my country." In his first speech as the new SNP leader, he added: "I will always seek, with respect and courtesy, to persuade people of the case for independence.
"All I ask of those who oppose that vision is they also act with the same courtesy and respect.
"That means independence is not a reason for them to refuse to work with us to build the economy, support jobs and improve the health service.
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