A banner reading “Trump Well Below Par” was revealed shortly after the US president arrived at the South Ayrshire hotel on Friday.
Police Scotland launched an investigation and said the incident was being treated as a breach of the air exclusion zone in place for the president’s two-night stay.
A 55-year-old man was arrested and charged on Sunday, police said.
He had been expected to appear at Ayr Sheriff Court on Monday.
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But a Crown Office spokesman said: “The procurator fiscal at Ayr received a report concerning a 55-year-old man and an alleged incident on Friday, 13 July.
“He was liberated from custody pending further inquiries and did not appear in court.”
Mr Trump and his wife Melania were staying at the Turnberry resort during a private leg of their visit to the UK, after the president had meetings with Prime Minister Theresa May and the Queen.