In a statement to the newspaper Ms Marney said: "I apologise unreservedly for the shocking language I used".
But one party politician said she should have been expelled, while a UKIP member of the European Parliament said it was time for the "inexperienced" leader to "go quietly".
Mr Oakden said: 'In light of messages that have been brought to my attention, I have taken the decision to immediately suspend her party membership pending an investigation.
Markle, who will Wednesday Harry in May, has spoken publicly and written about being biracial, including how it affected her acting career.
"Also it would be hypocritical to allow her to stay. because he said in the leadership election he was the candidate against racism and nazis so she has to go or he and @UKIP are doomed if we let this behaviour happen in the party", he added.More news: All the Nintendo Switch games from Nintendo Direct Mini
Party chairman Paul Oakden said he chose to suspend Ms Marney's party membership immediately after he was made aware of the messages.
Mr Bolton - who had been accused of covering up his new relationship during the leadership contest - faced demands to step down to limit the damage to the party.
Jo Marney, 25, whose relationship with the 54-year-old party chief emerged earlier this month, sent a series of messages to a friend in which she made offensive comments about Prince Harry's fiancee and black people, according the the Mail on Sunday.
Since these text messages were disclosed, Marney apologised and said that they were taken out of context.
"It appears to me that the lack of experience in politics from Henry has got the better of him", he said in a video statement, describing the last few months as "hell" for the party.
UKIP's London Assembly Member Peter Whittle called the remarks "disgraceful" and called for her expulsion.More news: General Motors (GM) Earns Daily Media Impact Score of 0.23
Bill Etheridge, a Ukip MEP, urged him: "Go now".
The Ukip leader confirmed that he had a "change in my relationship status" in recent weeks, although he denied reports that it had involved "a clandestine affair with a young lady who happens also to be a member of Ukip".
Federal Communications Commission Chairman Ajit Pai also took to social media to announce the panel would launch an investigation.
Mr Oakden declined to give his own views on Mr Bolton's situation, but said the party needs to be "behind our leader 100% in taking that battle (for Brexit) forward".
When a reply to one of her texts suggested her comments were racist, she added "so what?" and said she did not want other races invading her culture. "Yet I fully recognized the offense they have caused", she said.More news: USA has great respect for Africa - Embassy