An investigation by the Labour party into allegations of inappropriate behaviour by the politician Carl Sargeant, who died four days after being sacked as a Welsh government minister, has been dropped.

The party said it was no longer possible to take forward the investigation into Sargeant. The first minister of Wales, Carwyn Jones, also said he did not feel guilty that he had sacked Sargeant and insisted that he was confident he had acted properly.

Investigations have begun into how Jones handled the allegations against Sargeant, and whether news of his sacking was leaked.

A third inquiry is to focus on allegations of bullying within the Labour Welsh government.

The Labour general secretary, Iain McNicol, has written to Sargeant’s family solicitor to say the party’s investigation into him would not continue. In the letter, which has been seen by the Guardian, McNicol said: “It is no longer possible to take forward any investigation under our procedures and therefore the Labour party deems its investigation closed.”

McNicol said the Labour party had acted in accordance with its procedures. He wrote: “As you know our procedures require that after a complaint has been received the Labour party sends an agreed formal statement of the complainant to the respondent so that the respondent may revert.

“The Labour party received the initial complaint on Friday 3 November and was waiting for the formal statement to send to Mr Sargeant when the tragic news reached us of his passing.

“At no stage did the Labour party assume or confer any guilt on any individual … The Labour party rejects any liability for costs in this matter. We are confident we have acted in accordance with our procedures at all times.”

Sargeant was dismissed as cabinet secretary for communities and children on that same day, 3 November. He apparently took his own life at the family home in Connah’s Quay, north Wales, on 7 November. A complaint by Sargeant’s family and friends is that he did not know what allegations he was facing and had been left in limbo.

Speaking to the BBC, the Welsh first minister said he felt loss but not guilt over Sargeant’s death. “Carl was a good friend. I knew Carl for many, many, years. We never once argued. I’m confident that I did everything as I should have done. Could I have done something different? No. The only way I could have done it different would have been to have ignored the allegations.” There was no suggestion that Sargeant “was in a state that was vulnerable to do what he did”, he said.

Jones criticised the former local government minister Leighton Andrews, who said that the atmosphere in the Welsh government was toxic during the last assembly term. The first minister said Andrews was running “some kind of trial by Twitter” against the government.

Andrews replied: “I think the first minister needs to drop the personal attacks and smears. Our friend Carl Sargeant is dead. Attacking friends of Carl who are standing up for his family is not behaviour in keeping with the office of first minister.”