The SNP leader said the role of First Minister was an "immense privilege, but also an enormous responsibility" and she would fulfil the role “to the very best of my ability”.
She said: "In truth it’s the heavy sense of responsibility and duty that any incumbent in this office feels every single day.
“For me that’s never been more true than in these last 14 months or so. Our country is living through the most challenging and traumatic period that most of us have ever known.
“I have done my best to steer us through and I found out the hard way that in a situation like this there are very few good outcomes, just more or less harmful ones. Decisions necessary to save lives will cost jobs and businesses and effect our wellbeing in other ways and vice versa.
"I will continue to work every single day to make the best decisions possible and resolve never to shy away from tough calls.”
Ms Sturgeon said she hoped many of the SNP manifesto commitments would attract cross-party agreement, but that independence would be the issue that would require “open, informed and respectful discussion”.
"There is a clear mandate for a referendum within this term of Parliament,” she said.
"It is important in the interests of democracy for that to be acknowledged and respected. But it is also important that I exercise that mandate with responsibility, humility and only when the crisis of Covid has passed and I give that commitment today.
“How we achieve independence is as important as the outcome itself. It will determine the strength of the foundations on which we build.
"I am acutely aware that opinion on whether or not Scotland should be independent is evenly balanced. My decisions as First Minister for all of Scotland will always be mindful of that.
"That should apply to all of us. The views of those who do not support independence should not be ignored – you must not feel bludgeoned towards an outcome you have not been persuaded of – but by the same token the views of those who do support independence cannot be ignored either. We can't simply be told that there is no democratic route to seeking the future we want for our country.”
She added: “That takes us to a fundamental point. Unity of purpose for any country is an important condition of success. But unity is not achieved by pretending that genuine differences of opinion don’t exist or one side of a debate has no right to be heard.
“Unity is best achieved through open, informed and respectful discussion and agreeing the only way to resolve differences is through democratic means. Passions always run high on independence, but if we can strive for mutual understanding, empathy, to listen more than to talk, to respect the power of democracy, I believe we can and will find the right path, together, at the right time.”
Ms Sturgeon's nomination will now go forward to the Queen for approval, before she is sworn in at the Court of Session on Wednesday.
Mr Ross, who has returned to Holyrood as a Highlands and Islands MSP but is also remaining MP for Moray, was a late entrant to the First Minister election, after Scottish Lib Dems leader Willie Rennie previously said he would challenge Ms Sturgeon for the position to stop an “SNP cakewalk”.
In a speech to MSPs, Mr Ross, who won the 31 votes of Tory MSPs, said when he was younger he had always wanted to be a dairy farmer, and his likelihood of winning was equal to a cow being able to “jump over the moon”.
After Ms Sturgeon’s election, he congratulated her, but said: “We cannot ignore what has happened over the last 14 years of the SNP government here in Scotland. Our education system falling down international league tables.
“Drug deaths have more than doubled since 2014 to record highs. Our economy stagnated before the rest of the UK for a decade, ferries delayed or over budget.
“Violent crime rising, teacher numbers slashed, the patient treatment time guarantee broken 330,000 times. The list could go on.
“But we have to take Scotland forward. We cannot spend the next five years mired in the same stale debates and disagreements that consumed and held back the last Parliament.”
Scottish Labour MSPs abstained in the election of the First Minister. Party leader Anas Sarwar, who is self-isolating, said Ms Sturgeon now needed to focus on the Covid recovery.
"There are more than 300,000 Scots who don’t know if they have a job to go back to,” he said.
"We have suffered an economic crash harder and deeper than the banking crisis. Pupils have missed out on almost a year of school.
"Our NHS and social care systems, battered but not broken, need support, reform and investment. And in just months the eyes on the world will be on Glasgow as we host COP26.
"To meet these challenges – and many more – the First Minister must be bold.”
Mr Rennie, who received the four votes of his MSPs in the election, said Ms Sturgeon should not fill the vacant role in her Cabinet for a constitution secretary, but should appoint a minister for recovery.
“This parliament needs to focus on partnership and recovery,” he said. “On that basis there should be no place in this new government for a minister for a referendum.”
New Scottish Green MSP and her party’s co-leader, Lorna Slater, said she hoped a more diverse Parliament would mean a new commitment to cross-party working.
“Will we, as a more diverse group, have a more positive working culture?” she asked.