The force said the suspect was in custody for questioning after being arrested in St George’s Square in Luton on Tuesday.
The King was temporarily steered away from the crowds outside Luton Town Hall by his security staff shortly after arriving.
He then resumed shaking hands with members of the public after being moved to a different area.
Just last month, a man was detained by police after appearing to throw eggs at the King and Queen Consort during a walkabout at Micklegate Bar, York.
The King and Queen Consort were visiting Yorkshire where the monarch unveiled a statue of his mother, the late Elizabeth II.
The 23-year-old was heard shouting “this country was built on the blood of slaves” as he was being detained by around four police officers after throwing eggs at the monarch - none of which hit.
The King was visiting Luton to ride the new cable-drawn Dart passenger transit system. He was also set to visit Luton Town Hall and the newly-built Guru Nanak Gurdwara.
Charles looked relaxed as he met members of the public during his visit, just a day after the Duke of Sussex spoke of a “dirty game” in a trailer for his new Netflix series.
On Tuesday morning, Kensington Palace announced that Charles and the Queen Consort are to attend the Princess of Wales’s Christmas carol service on 15 December.
The King, Camilla, Kate and the Prince of Wales will gather with other members of the royal family for the Together At Christmas service at Westminster Abbey.
The show of family unity will come on the day the final three episodes of Harry and Meghan’s docuseries drop on Netflix.
Additional reporting from PA