Police Scotland said no-one was injured in the incident which is alleged to have happened on Friday night on Waverley Bridge.
The Fir Park club have released a statement on behalf of Robinson which denies any wrongdoing on the part of the Motherwell manager.
A Police Scotland statement said: “A 45-year-old man has been arrested and charged in connection with a disturbance on Waverley Bridge in Edinburgh around 7.45pm on Friday, 13 December.
“No one was injured. A report will be sent to the Procurator Fiscal.”
In a statement released by Motherwell to Sky Sports on Saturday night, Robinson said: “I want to make clear that I deny the charges against me and any wrongdoing on Friday.
“I am very upset by what has become a distressing situation for my partner, myself and our families.
“We had a lovely day in Edinburgh and were trying to get home in good time before the incident occurred.
“It is not possible to give a detailed account when a legal process is underway, which adds to our family anxiety.
“I am confident this will be found to be a misunderstanding once the legal process has run its course.”