The 54-year-old former soldier said attempts to remove him were a distraction from efforts to ensure the government was delivering Brexit, and warned he would act against those involved in “factional in-fighting”.
“I shall not be resigning as party leader,” he said in a statement.
Brexit was “the most pressing matter facing our country, and I am determined not to allow the national executive committee (NEC) to distract the party away from participating fully in the independence debate”.
The NEC, Ukip’s ruling committee, on Sunday declared a vote of no confidence in their leader, prompting an exodus of senior figures who demanded he quit.
It followed a scandal over racist messages sent by Bolton’s girlfriend about Prince Harry’s mixed race fiancee Meghan Markle.
Bolton, who left his wife last year, ended the relationship with 25-year-old party activist Jo Marney after her messages were published in a newspaper.
But assistant deputy leader Mike Hookem, a member of the European Parliament, was among the party members who said they could no longer serve under Bolton amid what he said was “an almost farcical scandal”.
“Ukip is now in the preposterous situation of the leader’s private life being of more interest than the party,” Hookem wrote in his resignation letter.
The NEC’s vote of no confidence opens the way for a full vote by party members in 28 days.
Bolton said that during that time, he would seek to unite the party to campaign on Brexit issues.
He will also seek major reform of the party’s governing body and rules, saying the NEC had lost his confidence and that of party members.
“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,” he said.
Echoing US President Donald Trump during his election campaign, Bolton added: “it is time to ‘Drain the Swamp’.”
Meanwhile former Ukip leader Nigel Farage dismissed speculation that he may be plotting to create a new political party to ensure a full break with the EU.
“No thoughts of a new party, whatever some may speculate,” Farage said. Asked if he would return to frontline politics, he said: “Doubt it.”
Without the charisma of Farage, a former metals trader who charmed right-wing media, Ukip has suffered.
His first successor, Diane James, resigned after just 18 days and her successor, Paul Nuttall, quit after Ukip won only 1.8% of votes in last June’s parliamentary election.
Under Farage, Ukip had won nearly 4mn votes — a share of 12.6% — in the 2015 election on its anti-EU platform, projecting it to the forefront of British politics even though it only managed to win one seat in parliament.
