But he acknowledged that the controversy over the events of last month had become a “distraction” and that he is standing down with “immediate effect” despite appeals from colleagues to continue.
“Yesterday I indicated to the Conservative group in the Welsh Parliament that I wished to resign, but they urged me to reflect further, and we agreed to meet again on Monday,” said Mr Davies.
“However, for the sake of my party, my health and my own conscience, I simply cannot continue in post.
“Therefore, I am stepping down as leader of the Welsh Conservatives in the Welsh Parliament with immediate effect.”
Tory chief whip Darren Millar said he is also stepping down from his frontbench role in the Senedd.
In his statement, Mr Davies said he is “truly sorry” for his actions on December 8 and the following day when further drinking reportedly occurred.
“They have damaged the trust and respect that I have built up over 14 years in the Welsh Parliament with my colleagues and the wider Conservative Party but, more importantly, with the people of Wales,” he said.
Mr Millar said he wanted to apologise to his family, colleagues and constituents for any “embarrassment” that media reports of what happened may have caused.
“While I am advised that I did not breach coronavirus regulations, I am very sorry for my actions, especially given the impact of the tough restrictions that people and businesses are enduring,” he said.
“I am co-operating fully with ongoing investigations and will continue to do so.”
The two men were said to have drunk alcohol with party’s chief of staff Paul Smith and former Welsh Government minister Alun Davies, who has already been suspended from Welsh Labour’s Senedd group.
“There was no drunk or disorderly behaviour. We did not have to be escorted out of the building as some reports have suggested,” said Mr Davies.
“What we did was to have some alcohol with a meal we heated up in a microwave, which was a couple of glasses of wine on the Tuesday and a beer on the Wednesday. I broke no actual Covid-19 regulations.”
Mr Millar said that on both evenings, he had served himself a pre-ordered, pre-prepared meal which was reheated in a microwave and which he ate while drinking an alcoholic beverage and discussing work matters with colleagues.
“Social distancing was maintained throughout these working dinners and it simply did not occur to either me, my colleagues or the catering team that the arrangements may have been inappropriate,” he said.
Additional reporting by PA Media.