"It is in the best interest of the Home Affairs Select Committee that its important work can be conducted without any distractions whatsoever," BBC quoted the Yemen-origin MP as saying.
"I am genuinely sorry that recent events make it impossible for this to happen if I remain chair."
At the weekend, the Sunday Mirror published pictures showing the Leicester MP with male sex workers in a flat in north London that he owns.
Illegal drugs were mentioned during a secretly recorded conversation.
According to the BBC, Conservative MP Andrew Bridgen said he would refer the matter to the Commons Standards commissioner and may also report Vaz to the police.
A married father-of-two, Vaz said he was referring the paper's allegations to his solicitor.
On Monday, Prime Minister Theresa May said the public must have confidence in its politicians, while Labour leader Jeremy Corbyn said it was a "private matter".
He was the parliament's longest-serving British Asian MP and has chaired the influential Home Affairs Select Committee since 2007.
Vaz was due to meet his colleagues on the House of Commons Home Affairs Committee later on Tuesday, but decided ahead of it to step down from the chairman role he has held for nine years.
In his statement he said: "The integrity of the Select Committee system matters to me. Those who hold others to account, must themselves be accountable.
"I am immeasurably proud of the work the committee has undertaken over the last nine years, and I am privileged to have been the longest serving chair of this committee.
"This is my decision, and mine alone, and my first consideration has been the effect of recent events on my family," he said.
Vaz said he had recommended that the former Conservative minister Tim Loughton takes over as chairman of the committee pending the full process of MPs electing his successor in the role.
"I would like to thank my fellow members of the committee, past and present, for their tremendous support. I would also like to thank the clerks of the House for the amazing work they have done to strengthen the select committee system, we are not quite on par with the US, but we are getting there."
Labour MP Chuka Umunna, who sits on the Home Affairs committee, in a tweet said that Vaz had done a "fine job" as chairman, but had made the right decision in stepping down.
