Mr Robertson, who has previously served as SNP Westminster leader, had been widely tipped as one of those who would run for the top job.
He said it was a “privilege and honour” that party colleagues wanted him to take on the job.
But he added: “As the father of two very young children, the time is not right for me and my family to take on such a huge commitment.”
Mr Robertson, who was elected to Holyrood in 2021 after losing his Westminster seat in 2017, said he would work with the next SNP leader to “deliver progressive policies and economic success for Scotland and help secure Scottish independence within the European Union”.
Confirmation that he will not stand comes as Kate Forbes, the Finance Secretary at Holyrood, is expected to announce her candidacy.
Five days after Ms Sturgeon dramatically quit the post, only two contenders for the job of SNP leader and first minister have emerged so far.
Scottish Health Secretary Humza Yousaf has already announced he will run for the post, as has former community safety minister Ash Regan, who resigned from her ministerial position in protest at the Scottish Government’s gender reform plans.
Business minister Ivan McKee said an announcement from the “hugely talented” Ms Forbes could be expected on Monday.
He told BBC Radio Scotland’s Good Morning Scotland programme: “Kate has some big decisions to make, of course, this happened while she was on maternity leave, she was not planning to come back until the beginning of April and as we all know the announcements of last week were a surprise to everyone.
“So Kate has had to assess her position, understand what can be done, but my understanding is that she has considered that and will be making an announcement later today and we shall see what she’s got to say.”
Mr McKee said he would be backing Ms Forbes if she decides to run.
He said: “I think Kate is a hugely talented individual, I’ve had the pleasure of working with her over the past seven years and seen first-hand what she is able to deliver, her focus on the things that are important to Scotland and her ability to engage with people on a very real and authentic level and persuade them of her case.
“And I think those talents, to be able to deliver excellent public services, a strong economy, a green economy for Scotland and to be able to persuade people of the case for the SNP and for independence, are exactly the assets we need in a leader.”
SNP depute leader Keith Brown and Deputy First Minister John Swinney have ruled themselves out of the race, as have Stephen Flynn, the SNP’s Westminster leader, Ukraine minister Neil Gray and the environment minister, Mairi McAllan.
Ms Forbes has long been seen as a potential successor to the First Minister.
She was thrust into the limelight by having to step in and deliver the Scottish budget at the 11th hour in 2020 after her predecessor, Derek Mackay, resigned in disgrace.
But the MSP for Skye, Lochaber and Badenoch, who is a Free Church of Scotland member, could be at odds with the wider party – and current policy – on some social issues, including the hotly-debated gender reforms blocked by the UK Government.
Candidates have until Friday to secure 100 nominations from at least 20 local branches to secure their place on the ballot, with the new SNP leader being announced on March 27.
