State police in OR said Thomas Wayne Markle Jr, 52, was visibly intoxicated when he was pulled over at about 1.30am on Friday in Josephine County, in southwestern Oregon.
The 52-year-old from Grants Pass was taken to a detox center.
The former Suits actress - who is now pregnant with her and Prince Harry's first child - is estranged from much of her family, including her brother, her dad, Thomas Snr, and her half-sister, Samantha.More news: Flu season poised to be milder than last year's
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Kensington Palace did not comment on the staffing changes.
It has been challenging for the popular new royal's bodyguards to accommodate her wish to be seen as "one of the people", sources told The Daily Mail on Sunday - as it was reported that her chief protection officer is leaving her position after only about six months on the job.
Meghan Markle had a very busy week, which included the announcement that she is receiving her first four royal patronages from Queen Elizabeth. "It's hopeful that if Meghan and Harry came and my father, Meghan and Harry could spend some time together with him and that would be good".
The woman, an inspector with the Metropolitan Police assigned to cover the British royal family, can not be named for security reasons, but had a 15-minutes-of-fame moment in Fiji in October when she leapt into action to remove Meghan from a market which became overcrowded and was deemed a safety threat.