The Tory MP made the statement for acting in a "threatening and intimidating manner" towards the complainants after he was unable to join a committee hearing due to technical problems.
A disciplinary report said the Shrewsbury and Atcham MP consumed a "significant amount of alcohol" on the day and phoned the manager of the committee staff while under the influence, behaviour that was "grossly unprofessional".
The incident occurred in April 2020, as Parliament was adapting to new remote working during the coronavirus lockdown.
Speaking in the Commons, Mr Kawczynski said: "I did not swear nor raise my voice but my behaviour led to two complaints. I have reflected on my behaviour, I accept it constituted bullying and as such was highly inexcusable.
WATCH MP's Commons apology:
"The circumstances were stressful for the staff who were assisting the committee and for me.
"I've apologised to them before and I apologise to them again and to the House unreservedly. I will never repeat such behaviour."
But before making the Commons statement, Mr Kawczynski told BBC Radio Shropshire he was only doing it because he had "no alternative" and would face further sanctions if he refused.
The Independent Expert Panel (IEP), which determines appeals and sanctions in bullying cases, ruled Mr Kawczynski should apologise in the House.
Sir Stephen Irwin, chairman of the IEP, said: "We accept that the circumstances which arose on April 27 2020 were difficult.
"But they were difficult for everyone. Whilst we fully grasp that the life of an MP can be highly pressurised, these responsibilities and stresses do not justify a loss of courtesy, an exaggerated sense of importance or entitlement, or bullying."
Parliamentary Commissioner for Standards Kathryn Stone concluded that Mr Kawczynski had acted in an "intimidatory and threatening manner" towards the staff - known as C1 and C2 in the report - and "abused his power" as an MP by making malicious claims about their poor performance.
The MP had been "repeatedly rude, aggressive and impatient" with the two complainants and other staff and made "critical and untruthful comments" on a WhatsApp group shared with other members of the committee.
C1 complained about the MP's behaviour in "several" phone calls and in the WhatsApp messages.
C2 also complained about phone calls and WhatsApp comments, with Mr Kawczynski's behaviour leaving the complainant feeling "anxious, upset, harassed and intimidated".
The report said the MP accepts that on several occasions on April 27 he let C1 and C2 and others know of his dissatisfaction but acknowledged that frustration, along with more "long-standing personal issues", may have had a "detrimental impact on the way in which he behaved", the IEP report said.
The IEP heard that Mr Kawczynski used "extreme, although not profane" language in his repeated calls to C2, with remarks such as "this is a farce" and "you are useless".
He threatened to make a formal complaint and referred to C2 as a member of the "snowflake generation".
The IEP said Mr Kawczynski had apologised to the complainants but "some concerns remain as to the sincerity of the apologies given to date".
Mr Kawczynski failed in an appeal against the recommended sanction that he should apologise in the House, claiming that he was under great pressure at the time due to Brexit and flooding in his constituency.
The appeal suggested the 6ft 9in MP was "very conspicuous" due to his height, making him a target for "extremely vicious" attacks from members of the public.
Boris Johnson believes it is "entirely right" for Mr Kawczynski to apologise, Downing Street said.
The Prime Minister's official spokesman said: "The Prime Minister is clear everyone working in Parliament deserves to be treated with dignity and respect, and there can be no place for bullying or abuse in Westminster or any workplace."
But ahead of his appearance in the Commons, Mr Kawczynski sought to play down the incident, telling BBC Radio Shropshire: "We all have bad days - but in exasperation at the end of the day I sent a message on the WhatsApp group to my fellow MPs on the committee, and the clerks are also on the communal WhatsApp group and I said 'I'm terribly sorry but I still haven't been able to log on all day, I feel badly let down by the clerks and I'm going to resign from the committee'.
"That has resulted in a 15-month investigation into me by the House of Commons."
He added: "I have no alternative but to apologise. I must apologise because if I don't apologise then I risk the option of being sanctioned further - namely being suspended from the House of Commons or expelled from the House of Commons."
Mr Kawczynski said he is now co-writing a book on "emotional intelligence and politics".