The First Minister has been slammed for the handling of inappropriate behaviour allegations against Mr Sargeant, who is believed to have taken his own life.
Mr Sargeant, 49, was still unaware of the allegations against him at the time of his death on Tuesday.
This afternoon, Mr Jones was locked in discussions about the tragedy with Labour AMs in Cardiff Bay.
And, in a statement to the press, he paid tribute to the politician but made no mention of resigning.
He told reporters: “I probably did all that I could to make sure everything was being done by the book.
“I had no alternative but to take the actions that I did and I hope people will understand that.”
The First Minister said the events were “the darkest days” the Welsh Assembly had ever faced.
Mr Sargeant was facing allegations of “unwanted attention, inappropriate touching or groping” and was sacked as a Welsh Government minister and suspended from the Labour Party on Friday by Mr Jones.
But the Alyn and Deeside AM’s family said he had been denied “natural justice” because he was unaware of the details of the allegations against him at the time of his death.
His family said Mr Sargeant’s distress at being unable to defend himself properly meant he was not afforded “common courtesy, decency or natural justice”.
Bernie Attridge, deputy leader of Flintshire County Council and a councillor in Mr Sargeant’s home town, Connah’s Quay, has demanded the First Minister’s resignation.
He said he was angry about the way his friend had been treated.
And, in a message to Mr Jones, Mr Attridge said: “I call on you to do the right thing and resign.
“The way you have treated Carl is unforgivable, you make me sick.”
It is believed Mr Sargeant took his own life four days after being sacked from his role as communities and children secretary in the Welsh Government.
Labour AM Jenny Rathbone also raised concerns about the situation and said Mr Sargeant’s family were “devastated”.
She said: “It is a basic fairness that you have got to know what the accusations are against you in order to be able to respond.
“But that doesn’t mean to say we don’t need to take allegations seriously, we obviously do.”
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