Bolton said a new leadership election to replace him would be "unviable" for Ukip's finances and could finish it as a party.
Neil Hamilton, Ukip's leader in Wales, said Mr Bolton should "seek psychological help".
He has not ruled out reuniting with her in the future but made it clear he did not agree with her comments.
Bolton is the pro-Brexit party's third leader since high-profile politician Nigel Farage stepped down in November 2016.
Mr Bolton has faced repeated calls to quit from within his own party, despite breaking off his relationship with 25-year-old Jo Marney earlier this week.
But he said she had received death threats and he wanted to support her in "rebuilding her life".
Mr Bolton warned that another leadership contest could spell the end for the party, which is believed to be heavily in debt, and said calls for his resignation were part of a "coup" attempt by a rival faction in the party.
"If the NEC decides to go down the road of months of further in-fighting and further negative media scrutiny by deciding to pass a vote of no confidence in me, the reality is the party is probably over".
Greeks rally against use of 'Macedonia' in FYROM name dispute
The crowd is chanting "Hellas - Macedonia", "No to the double name", "Not a new betrayal" and are singing Macedonia themed songs. The 1992 rally was backed by the Greek Orthodox Church which, however, distanced itself from Sunday's demonstration, as a whole.
Asked if he was going to quit, the party leader replied: 'No I'm not, no'.
Mr Bolton said previously his "romantic" relationship with Ms Marney was over following an outcry sparked about a series of shocking comments she made.
Mr Bolton admitted that is private life "has been a bit of a mess" and plunged has party into a fresh crisis. "I need to sort that out, of course".
Farage had been expected to announce a new party as long ago as September, when it appeared Anne Marie Waters, a UKIP politician known for virulently anti-Islamic views, would win UKIP's leadership vote.
"The vote was carried unanimously with the exception of the Leader".
This decision will automatically trigger an EGM (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.
But a Ukip source told HuffPost UK the coffers are sparse, and spending money on sending ballots and election literature to nearly 25,000 members was an expense the party could ill afford.
Bolton's position has become increasingly precarious since it emerged he had left his wife for Ukip activist Jo Marney - a move which some in the party felt undermined his "family man" image.