Bill Hedden, 59, is recovering at the Medical University of South Carolina from the attack in Lake Moultrie near Moncks Corner on Sept. 16.
The arm was later recovered from the alligator, which was killed. The injuries were so severe that doctors did not attempt to reattach the limb.
Mr. Hedden has not met with reporters but his son, Brian, 34, told reporters Monday that his father was always cautious before getting in the water.
"Bill recognizes that alligators are a potential threat anywhere on the lake or anywhere around the Lowcountry," the younger Mr. Hedden said. "Through 25 years of comings and goings to this area, he also felt the risk in this area was minimal."
The elder Mr. Hedden, a retired U.S. Navy master chief from Summerville, S.C., swam in the lake three times a week.
"It's just general exercise," his son said. "He usually does swim in a different area of the lake, but at that point in the afternoon it was really windy and really choppy and he shopped for a little more secluded, calm area."
Bill Hedden was only in the water for about five minutes before the attack occurred. Making his way back to shore, nurses who happened to be at the recreation area applied pressure to the wound, stopping the flow of blood.
"To me, these bystanders are the heroes of this story. Without their prompt action, Mr. Hedden may very well have bled to death," said Dr. Doug Norcross, an MUSC trauma surgeon who treated Mr. Hedden.
Brian Hedden said his father is in good spirits and will soon be released, although doctors have not determined exactly when he will be able to go home.
He said his father is amazed by the intense media interest.
"I don't think he has a grasp on it - just how big it got in the news media," he said. "He really thinks of himself as a normal person who had an extreme circumstance."