He told Sky News he was "appalled" when he saw her comments, ending his relationship with the former model "after many hours of difficult conversation".
He said the texts were made "in the heat of the moment" and "not meant for the public eye", but were "not defensible in any way".
In a series of text messages sent to a friend, Ms Marney said that the Prince's "black American" fiancee will "taint" the Royal family with "her seed" and pave the way for a "black king", according to a report by the Mail on Sunday.
The 25-year-old, who describes herself as a model, actor, journalist and Brexiteer on her Twitter profile, also reportedly said Ms Markle had a "tiny brain" and that black people were ugly.
She has since been suspended from the party.
While he said the chairman was "right" to suspend Ms Marney, he would not say whether he thought she should be thrown out of the party, saying the decision was "nothing to do with him".
Responding to widespread calls for him to stand down as UKIP leader, he said the party "can't afford a leadership change", insisting "I've got a great amount of support".
He said his critics are using the incident to attack him, but said members should "work towards uniting the party instead of dividing it".
He told Sky News: "I'm pushing forward with the agenda of reforming the internal communication, finances and policy making of the party.
"All stuff that needs to be done to allow the party to project its politics into the Brexit debate and the shaping of the country post-Brexit.
"This is all critical stuff and we can't afford a leadership change at the moment."
Former UKIP leadership candidate Ben Walker has called for Mr Bolton's resignation, accusing him of having "deeply flawed judgement", claiming he is damaging the party.
UKIP's National Executive Committee will meet to discuss Mr Bolton's future as leader of the party this Sunday.
Sky News Political Correspondent Lewis Goodall said it is likely there will be a vote of no confidence at the meeting, with one of his sources telling him that Mr Bolton "has few allies and members are leaving in droves".
A senior UKIP figure told him: "The chances of Henry Bolton surviving as UKIP leader are extremely small."
If Mr Bolton, who was elected party leader in September, were to leave the party it would force UKIP to elect its fifth leader in 18 months.
Mr Bolton, 54, announced his relationship with Ms Marney in a letter to UKIP supporters earlier this month, but denied a "clandestine affair".
He left his wife Tatiana, 42, with whom he has two children, before going public with the relationship.
He said he met Ms Marney at a Boxing Day lunch where he spoke to her "for about five minutes".
Talking about her political opinions he said, "we have discussed in the past, immigration and integration issues and she has never expressed her views in such a way as she did in those messages."
He said Ms Marney is "utterly distraught" over the revelations.
She has since apologised "unreservedly" for her comments, but insists her opinions have "been taken out of context".
Mr Bolton said that both he and Ms Marney now "need to put out lives back together after this".
