Cpl. Wassef Ali Hassoun "has had a full night's sleep and some breakfast. He's up and awake now, and he has been given telephone cards and access to a phone," said Maj. Tim Keefe, who visited Hassoun on Saturday morning.
Hassoun is expected to remain at Landstuhl Regional Medical Center, where he is being debriefed and evaluated by doctors, until early next week, hospital spokeswoman Marie Shaw said.
He then will return to his home unit in Camp Lejeune, N.C., Keefe said.
Hassoun is a Lebanese-born Muslim who was working in Iraq as an Arabic translator, Keefe added.
Hassoun, of West Jordan, Utah, arrived in Germany on Friday night from Lebanon, where he turned up at the U.S. Embassy in Beirut on Thursday. The 24-year-old had been missing since June 20 from his base near the troubled Iraqi city of Fallujah.
During the last three weeks, conflicting reports emerged about Hassoun's fate. Some said he was beheaded.
On June 27, Arab television showed a videotape of Hassoun with his eyes covered by a white blindfold and a sword hanging over his head.
The Navy is now investigating whether the kidnapping might have been a ruse.
Hassoun is out of danger, but he is not out of trouble, reports CBS News National Security Correspondent David Martin.
Technically in time of war, desertion is punishable by death, but nobody's talking about that right now," said Martin, at the Pentagon. "Everybody's just trying to figure out what the heck happened here."
Doctors who evaluated Hassoun on his arrival at Landstuhl said he was in excellent physical condition but had lost about 20 pounds.
Military investigators are not expected to question Hassoun while he is hospitalized, Keefe said.
"At this point he is a patient; we have to learn the facts before anything is done," Keefe said.
"I don't think it's going to come out for quite a while, and that could mean several days or several weeks."