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.
Gerry Adams has denied claims in a BBC Spotlight programme that he sanctioned the murder of an IRA informer.