In a statement to the newspaper, Ms Marney apologised "unreservedly" for the "shocking language" she used in the messages.
In the series of alleged vile slurs, Henry Bolton's 25-year-old lover Jo Marney said Ms Markle's "seed" would "taint the Royal Family".
Ms Marney wrote that Ms Markle, a United States actress whose mother is black and father is white, will "taint" the royal family, the paper quoted one of her messages as saying.
Mr Etheridge said he had not voted for Mr Bolton in the leadership election but had been prepared to support him when he won it.
"I think it is very clear that Henry is increasingly in a position where he's got some hard decisions to make", he said on BBC's Sunday Politics. "Go quietly and leave us to deal with what's left".
Peter Whittle, leader of UKIP's delegation in the London Assembly, also called for Marney to be "expelled altogether" for the "disgraceful remarks".
The party chairman, Paul Oakden, said the leader had "some hard decisions" to make.
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The report of Ms Marney's use of highly offensive language about people from different ethnic backgrounds comes as Mr Bolton faces an investigation into his controversial private life by senior party officials.
She is believed to have sent the messages to a male friend.
UKIP's national executive committee will meet next Sunday to decide what to do, unless Bolton resigns as leader before then.
The pair's relationship became public earlier this month after the Ukip leader confirmed that he had changed his "relationship status" in recent weeks.
He claimed that his marriage to Ms Smurova was already over when he met Miss Marney, and has said: 'I have no doubt I'm going to come out of this, and Jo and I are going to come out of this as a couple, stronger than we went into it'.
She said: "The opinions I expressed were deliberately exaggerated in order to make a point and have, to an extent, been taken out of context. Yet I fully recognise the offence they have caused", she added.
During the exchange, Ms Marney - who is from Kent and describes herself as an actor and music journalist - also complains that there is "an immigration problem", saying: "This is Britain".