That plea, sent to 20 Navy personnel, became public and angered Mr. Modly, which led to his decision to remove the captain from his post.

Admiral Gilday indicated last week that he was open to reinstating Captain Crozier once the preliminary investigation was completed. “I am taking no options off the table as I review that investigation,” he told reporters. “I think that is my responsibility.”

Any decision to reinstate Captain Crozier would come with its own problems. Navy officials remain unhappy that the captain sent an unclassified letter seeking help to so many people, instead of relying on his chain of command. For the Navy to reinstate him, Admiral Gilday would have to determine that Captain Crozier’s superiors were not adequately responsive to requests for help before the letter had been sent, Navy officials said.

Admiral Gilday already has the findings of an initial investigation into the Roosevelt case. But that investigation was conducted by Adm. Robert P. Burke, the Navy’s second-highest admiral, who was involved in the situation aboard the Roosevelt. As part of the investigation, Admiral Burke called the senior medical officer aboard the ship and criticized the doctor, saying he had failed as a leader, according to crew members.

In the run-up to Captain Crozier’s letter, the ship’s medical staff and the captain advocated swift, decisive action, while Captain Crozier’s immediate boss, Rear Adm. Stuart P. Baker, countered that less drastic measures would still protect the crew and leave the Roosevelt in operation.

Admiral Gilday told reporters this month that the investigation would focus in part on why Captain Crozier, a Naval Academy graduate with nearly 30 years of service, felt compelled to send his four-page letter outside normal communications channels and whether that illustrated a breakdown in communications with his chain of command, particularly with Admiral Baker. The Navy has said Captain Crozier did not copy Admiral Baker on his letter.

Before the results are made public, Admiral Gilday will consult with the new acting Navy secretary, James E. McPherson, as well as with Defense Secretary Mark T. Esper and General Milley.