The first minister said images of Aylan Kurdi being carried from the sea had "touched our hearts".
Ms Sturgeon is to convene a migrant crisis summit on Friday with councils and humanitarian agencies.
Prime Minister David Cameron has come under increasing pressure for Britain to accept more refugees.
Scottish Conservative leader Ruth Davidson is among those who have urged Mr Cameron to do more to tackle the refugee crisis in Europe.
But Immigration Minister James Brokenshire said accepting more refugees was not the answer.
Speaking at First Minister's Questions, Ms Sturgeon said: "As first minister I pledge that I will ensure Scotland will do everything possible to help in this refugee crisis.
"I will be far from the only person reduced to tears last night at the picture of a little boy washed up on a beach.
"That wee boy has touched our hearts but his is not an isolated tragedy.
"He and thousands like him whose lives are at risk is not someone else's responsibly; they are the responsibility of all of us.
"So yes I am angry, I am very angry at the walk-on-by attitude of the UK government and I implore David Cameron to change his position and change it today.
"I pledge as first minister of this country that we stand ready to offer sanctuary to refugees that need our help."
Scottish Labour leader Kezia Dugdale also criticised Mr Cameron's response to the crisis.
She urged the first minister to "convene an urgent meeting with Scotland's council leaders, party leaders and people in this parliament and other relevant government agencies so that Scotland can speak with one voice and match our compassion with the action we are all willing to take".
Ms Sturgeon said she had already set up a summit, which will bring together humanitarian organisations, including the Scottish Refugee Council, leaders of councils, civic organisations and religious leaders.
Scottish Liberal Democrat leader Willie Rennie said he would accept the invitation to attend the summit.
Ms Davidson said the UK government must act "closer to home" and go beyond donating aid to Syria, adding: "This is not an immigration issue - it's a humanitarian crisis."
She added: "I call on David Cameron to lead our government's response and accept more refugees."

Ceredigion-based Lewis Coaches, which runs buses from Aberystwyth to Cardiff via Carmarthen, is to shut on Friday.
The Confederation of Passenger Transport Cymru said 40 jobs were at risk.
Ceredigion council said it was working to continue some services.
The authority said plans were already in place to see the on-demand Bwcabus service and the 588 and 585 routes from Aberystwyth University to Lampeter taken over by other firms by Saturday.
But the council said it understood the 701 Aberystwyth - Carmarthen - Cardiff service would end on Friday and it did not know if another operator would take it on.
Lewis Coaches is the third bus company in Wales to collapse in the past two months, with 300 jobs lost when Wrexham-based GHA Coaches went into administration in July and Pembrokeshire firm Silcox Coaches, ceasing trading after 134 years in June.
Nicola Sturgeon has told how she was reduced to tears by photos of a dead Syrian boy washed up on a Turkish beach.
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Some bus services in mid Wales facing the axe because the firm which runs them went into administration could be maintained, a council has said.