Morison, 67, says the decision has "vindicated" reporting on Rohingya Muslim boat people in his online news service, Phuketwan.
The Royal Thai Navy had for two years pursued charges that could have seen Morison, a former senior Age editor, and Chutima, a reporter on Phuketwan, jailed for up to seven years.
The charges related to one paragraph referring to "Thai naval forces" allegedly having knowledge of the smuggling of Rohingya, who have been described by the United Nations as among the world's most persecuted people.
Phuketwan reproduced the paragraph in 2013 from a Reuters' series of stories that won the international news agency a Pulitzer Prize.
The navy did not charge Reuters.
The decision not to appeal a Phuket court's verdict on September 1 came amid reports that another court has issued arrest warrants for four senior Thai military officers on charges of human trafficking, including a naval commander.
Morison has put his Phuket-based company up for sale, saying defending the charges had been a huge financial and personal burden.
"I guess we derive a great deal of satisfaction from being proved right, even if it has cost us quite a bit to be true to that," he said.
"We hope Phuketwan will continue but we're not likely to be involved."