Daniel Kawczynski issued a personal apology in the chamber over the matter and insisted he is committed to "learning from the mistakes" he has made.
The one-day suspension has come into effect on Thursday after MPs approved the recommended punishment.
The Commons Standards Committee recommended action be taken after media interviews Mr Kawczynski gave appeared to call into doubt the sincerity of his earlier apology.
The MP for Shrewsbury and Atcham made an apology in June last year after he was found to have breached rules on bullying and harassment following a complaint by Commons staff.
However, the same day he told an interviewer from BBC Radio Shropshire: "I have no alternative but to apologise because if I don’t apologise then I risk the option of being sanctioned further."
An investigation by Parliamentary Commissioner for Standards Kathryn Stone found he also breached confidentiality rules by identifying the complainants through their job titles.
Mr Kawczynski’s conduct was particularly serious as it risked undermining the credibility of the independent complaints and grievances scheme for Commons staff which has only recently been established, the committee said in its report.
Speaking in the Commons on Wednesday, Mr Kawczynski said: "I acknowledge that in speaking to journalists and the radio I undermined the sincerity of the apology I gave to the House on June 14 2020.
"I’m sorry that my conduct will have had a further harmful effect on the complainants and that it may have diminished public confidence in the process.
"I will be sending a written apology to the parliamentary commissioner for standards, the chair of the Independent Expert Panel, and the original complainants.
"I am conscious that my conduct may have merited suspension from the service of the House for a longer period and appreciate the committee’s consideration of the difficulties I was experiencing in my personal life at the time and the mental health issues I explained to them."
He adds: "I accept that speaking out in the way I did to the media had a detrimental effect on the House’s conduct policy by undermining the integrity of the complaints process and I deeply regret my actions.
"I am committed to learning from the mistakes I have made and to work on my personal development, especially in my communication with others in every interaction that I have. I hope others will learn from my experience and I’d be happy to share what I have learnt with others."
For Labour, shadow Commons leader Thangam Debbonaire said: "It appeared to me that his apology was genuine and sincere."
She added: "I wish the honourable gentleman well in his process of change."
Mr Kawczynski was a parliamentary neighbour and ally of Owen Paterson, who quit last year as MP for North Shropshire after a botched attempt by Boris Johnson to get him off the hook when he was said to have broken rules banning paid lobbying by MPs.