In the programme, a man who said he too was a former IRA and Sinn Féin member turned British agent, alleged that the Sinn Féin leader approved the killing of Denis Donaldson in 2006.
The former spy cannot be identified because of fears about his safety.
Mr Donaldson was shot dead months after admitting that he had been an MI5 agent for more than 20 years.
Three years later, the Real IRA said it was responsible.
On Wednesday, Mr Adams said he "specifically and categorically" denied the allegations, describing them as "part of the British security agencies ongoing attempts to smear republicans and cover up their own actions".
He said he would consider taking legal action against the BBC.
"I've taken a number of cases against various periodicals, I got a number of corrections and apologies from a number of outlets," said Mr Adams.
"If my legal advisers give me the type of guidance that I require, I will sue them. I'm not reluctant to sue.
"That the BBC would broadcast unsubstantiated allegations from an anonymous person, a self-confessed agent, about me, I think it's very, very low journalism indeed."
Mr Adams said there was a need for all agencies to "fully co-operate with the ombudsman's investigation into the role of the police in the events that led to the killing of Denis Donaldson and for the Gardaí to expedite their investigation to bring those responsible to justice."
The Sinn Féin leader said he would "continue to support the family of Mr Donaldson to achieve truth and justice".
Mr Donaldson had worked for Sinn Féin as a senior administrator at Stormont. He was killed at a remote Donegal cottage in 2006.
What Spotlight was told
Agent: I know from my experience in the IRA that murders have to be approved by the leadership. They have to be given approval by the leadership of the IRA and the military leadership of the IRA.
Presenter: Who are you specifically referring to?
Agent: Gerry Adams, he gives the final say.
The informer told Spotlight that murders had to be approved by the political and military leadership of the IRA.
Days after Mr Donaldson's killing, the IRA said it was not involved in Mr Donaldson's death.
But security sources have told Spotlight that intelligence received following the killing contradicted the IRA's denial.
You can watch BBC Spotlight: Spy in the IRA now on BBC iPlayer.

The Jersey hoard was found in a field by metal detectorists Richard Miles and Reg Meade in 2012.
The leather purse is about the size of an average palm and is described as an important find by senior conservator Neil Mahrer.
He said they have so far removed some 40,000 coins and this was the first bag they had come across.
Mr Mahrer said: "It is made of leather, it looks like it is either made of one sheet or bootlace strands.
"It is about 8cm across and 4cm deep. It is joined at one side to two rings that could have been for fixing it to a belt."
The team from Jersey Heritage have been working for the past 18 months on removing the coins one at a time. There are an estimated 70,000 coins in the hoard.
Mr Mahrer said: "The really important thing that might change our ideas is when we get to look at the coins themselves as we don't know the significance of the purse to the people who buried the hoard."
He said they would be removing all the coins surrounding the purse so they could then remove it with the earth in a single block.
The previous major find from the hoard was a selection of gold jewellery.
Mr Mahrer said: "Since removing the gold torques we thought we were in a coin-rich areas but seeing this it looks like we're in a rich seem of surprises again.
"Finding a purse where the whole thing is intact with the coins inside is pretty rare."
Gerry Adams has denied claims in a BBC Spotlight programme that he sanctioned the murder of an IRA informer.
[NEXT_CONCEPT]
A rare purse has been found in the middle of one of the world's largest hoards of celtic coins.