"I have written a letter to the Iranian people to apologise for us entering their waters,'' it said. The letter also asks Turney's parents in Britain to look after her 3-year-old daughter, Molly, and her husband, Adam.
Turney was the only detainee to be shown speaking, giving her name and saying she had been in the navy for nine years.
"Obviously we trespassed into their waters," she said at one point.
"They were very friendly and very hospitable, very thoughtful, nice people. They explained to us why we've been arrested. There was no harm, no aggression."
Margaret Beckett, the British foreign secretary, said after the broadcast: "I am very concerned about these pictures and any indication of pressure on or coercion of our personnel who were carrying out a routine operation in accordance with international law."
Des Browne, the British defence minister, said it was "completely unacceptable to parade our people in this way".
The government maintains that the sailors had been seized in Iraqi waters.
After the broadcast, Manouchehr Mottaki, Iran's foreign minister to the UK, said the British government must accept the sailors were arrested in Iranian waters, but said Turney would be freed "as soon as possible".
Mottaki had said earlier that Turney would be freed on Wednesday or Thursday.
The British interior ministry said it had no confirmation of any pending release and demanded access to the group, detained at a time of high tension between Iran and Western countries over Tehran's nuclear programme.
Browne had earlier said global positioning data showed the British sailors and marines were 1.7 nautical miles within Iraqi waters when they were captured by Iranian gunboats near the waterway that separates Iran and Iraq.
But Iran's embassy in London said the British sailors and marines were 0.5 km inside Iranian waters at the time.
It said it had given that data to Britain and was confident that the countries could resolve the issue through co-operation.
US naval exercises
With the US conducting naval exercises in the Gulf, the rising tension rattled global markets.
Oil prices jumped by $5 overnight to more than $68 a barrel before they settled back to around $64. Gold jumped to a four-week high.
For the first time since the US invasion of Iraq in 2003, a second US aircraft carrier, the John C Stennis, arrived in the Gulf for previously scheduled naval war games.
But the US and Iran played down the naval exercises, with Washington saying they did not escalate tensions and a headline on Iranian state television screens reading: "Iran: 'No concern about Pentagon's war games in the Persian Gulf'."
Tony Blair, the British prime minister, said the detention of the sailors was "wrong and illegal" and told parliament that it was time to "ratchet up" the pressure on Tehran.
The European Union extended its "absolute support and solidarity" to Britain while the US called Iran's behaviour "reprehensible".
In a similar incident, Iran freed eight British service members after holding them for three days in 2004.