It comes after ano-confidence motion by Ukip's national executive committee (NEC).
He has also said he has ended the "romantic element" of his relationship with girlfriend Jo Marney after she sent texts saying Prince Harry's fiancee Meghan Markle would "taint" the Royal Family, leading to accusations of racism.
Several members of the UKIP leadership quit in protest, including deputy leader Margot Parker and the party's immigration, trade and education spokesmen.
Calling for "cohesion" in Ukip and rather than focusing on the leadership debate, Mr Bolton urged the Party to "concentrate on such matters as the local government election campaign and the necessity of mobilising our efforts to ensure the government delivers true independence in all areas of government and administration when we leave the European Union".
He said: "I respect the next steps in the constitutional process and will therefore not be resigning as Party leader".
"The vote was carried unanimously, with the exception of the leader", the UKIP said after yesterdays national executive committee vote.
Henry Bolton was announced as Ukip's new leader in 2017Who is Henry Bolton?
Mr Bolton continued that Ukip's ruling body "as presently constituted, is unfit for goal and has severely handicapped the Party's progress and political delivery for some years, as all recent UKIP leaders will attest".
While Peter Jewell resigned as Ukip's justice spokesman, and Julia Reid as environment spokeswoman, just hours after Mr Bolton defied calls to quit.
'This decision will automatically trigger an emergency general meeting of the party, to allow the membership of Ukip the democratic opportunity to decide to endorse or reject that vote of no confidence'.
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The departure of charismatic leader plunged party into a freaky succession of internal fights that, on one occasion, came to hands, with a candidate hospitalized after being knocked down in European Parliament's quarters for a punch by a fellow of rows.
UKIP chairman Paul Oakden tried to convince Bolton to resign at a meeting of the party's top officials over the weekend but was unsuccessful, Politics Home reported.
"The NEC requires significant and urgent reform".
"It is now time to put an end to the factional in-fighting within the Party and to remove those who have been a part of that".
The UKIP leader was being quizzed by one of his predecessors, Nigel Farage, who suggested the saga was becoming "cringe-making".
"In a single Phrase, it is time to "Drain the Swamp".
Party members will now be given a vote on whether the former Army officer should remain in post.
UKIP is also struggling financially.
Its popularity prompted Cameron, who once dismissed the party as being full of "fruitcakes, loonies and closet racists", to hold the referendum and played a major role in securing a vote to leave the EU.