Gary accused the company of gross negligence and putting profit ahead of the lives of its employees by deciding to sail into the oncoming storm.
"The ship should have never left dock. The ship was not seaworthy," Gray said.
"We hope to get to the bottom of this," he added, saying more lawsuits would follow on behalf of relatives of other crew members. "We're at war now," Gary said.
The company did not immediately respond to a request for comment. Tote executives have previously said the captain sailed with a sound plan and blamed the sinking on engine failure.
Jordan, 33, of Jacksonville, worked on the ship for 13 years as a cook and at other jobs, his family told the Jacksonville Times-Union.
The family told the newspaper the company made the wrong decision allowing the ship to set sail on its final voyage as Hurricane Joaquin was brewing in its path.
The 790-foot container ship left Jacksonville on a weekly cargo run to Puerto Rico on the evening of Sept. 29. It was last heard from on the morning of Oct. 1 when the captain communicated that the ship had taken on water, was listing at 15 degrees and had lost propulsion. Its last known position was close to the eye of the storm battling 50 foot waves and winds over 100 miles per hour.
The U.S. Coast Guard called off a search and rescue mission last week after finding only one body amid debris from the ship.
A National Transportation Safety Board investigation is under way and is coordinating a salvage team to retrieve the ship's voyage data recorder.