Carter said the sailors apparently did not radio in to tell their commanders they were off-course before encountering the Iranians.
Iran contends the sailors were attempting to snoop on a revolutionary base located on Farsi Island in the Arabian Gulf.
Iran determined that the breach of its waters was not an act of espionage but instead was an accident due to the navigation equipment on the boats being broken.
Iran state television also released images of the sailors in captivity Wednesday.
Screen shot from Iranian media report on detention of us sailors.
Iran held the 10 sailors-nine male and one female-and their two Navy boats in custody overnight Tuesday, accusing them of drifting into Iranian territorial waters while moving between Kuwait and Bahrain in the Persian Gulf.
"I'm pleased that 10 U.S. Navy sailors have departed Iran and are now back in U.S. hands", U.S. Defense Secretary Ash Cater said in a statement.
A USA military official said: "Clearly this staged video exhibits a sailor making an apology in an unknown context as an effort to defuse a tense situation and protect his crew".
While the sailors appear to be fine physically they are "tired and upset" about what happened, according to the official.
It is not yet clear if Lt. Nartker was directly ordered to make an apology in the video, the official said, CNN reported. The sight of Iran and the USA resolving a cross-border incident the way any two normal countries would must certainly be a hopeful sign for long-term improvements in their relationship.
The day of the incident, USA officials had said mechanical problems had caused the boats to stray into Iranian waters.
US military officials declined to provide any new detail on the timeline for what occurred.
State Department spokesman Mark Toner said the video was being assessed and the initial conclusion was that the sailors were treated well.
Secretary of State John Kerry praised Iran for quickly resolving the matter.
At a presidential campaign rally in the US, Republican frontrunner Donald Trump, who accuses President Barack Obama of being weak on foreign policy, described the incident as "an indication of where the hell we're going".
Kroenke heralds Rams' return after "long road" back to LA
The blue-and-yellow frenzy of pregame tailgating was more energetic than anything I'd ever seen outside a Rams game. The Rams will maintain temporary offices and a practice facility at the team's current Rams Park facility in St.
