Eight months after losing his Westminster constituency, Mr Robertson announced he was giving up his post within the party with immediate effect.
He said he was “tremendously honoured” to have served as Nicola Sturgeon’s number two in the party, telling the Scottish First Minister he valued her “friendship, professional and personal support”.
Mr Robertson lost the Moray constituency, which he had represented since 2001, to the Conservatives in last June’s snap general election.
In his letter to Ms Sturgeon, the former MP said: “As we have discussed recently, I believe I am no longer able to fully discharge my mandate, which was to partner you as Westminster SNP leader and as a parliamentarian representing a rural constituency.
“While it would be my greatest privilege to continue as depute leader, I know you understand that I have to focus now on pursuing new career opportunities.
“It is for that reason that I believe it is for others to step forward to seek the support of SNP members for the honour to serve as depute leader of Scotland’s largest political party.”