Labour party MP Vaz, in his capacity as chair of the House of Commons Home Affairs committee, said the decision was "a big shock" and announced that Dame Lowell Goddard has been asked to appear before British MPs and explain whyshe resigned as chair of the independent inquiry.
"I've written to her today to ask her to come before the committee when we return at the end of August and share with us her thoughts about the setting up of this inquiry and why she resigned, and where she thinks we could go," Vaz said.
Goddard has not yet given full reasons, but she said conducting such an inquiry was "not an easy task".
In a statement, she said she was "confident there have been achievements and some very real gains for victims and survivors of institutional child sexual abuse in getting their voices heard".
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The judge said she took on the role because of her "relevant experience and track record in the area" but admitted that running the inquiry had been difficult.
"Compounding the many difficulties was its legacy of failure which has been very hard to shake off and with hindsight it would have been better to have started completely afresh," she said.
UK home secretary Amber Rudd said the inquiry would continue "without delay" in the absence of a new chair.
The inquiry was announced in July 2014 to examine historic claims of child sexual abuse made against public and private institutions.
It had a budget of 17.9 million pounds in its first year, funded by the UK Home Office, with staffing-related costs accounting for 41 per cent of the total.
The original chair, Baroness Butler-Sloss, stood down just a week after being appointed amid questions over the role played by her late brother, Lord Havers, who was attorney general in the 1980s.