Speaking at a press conference after his release from custody after four days of questioning about the killing of Belfast mother-of-ten Jean McConville, the Sinn Fein president said he was "concerned about the timing" when the Police Service of Northern Ireland (PSNI) contacted his solicitor on Monday afternoon.
But Mr Adams, who prompted cheers from those in the room before he made his statement, asserted his support for the PSNI, saying: "I want to make it clear that I support the PSNI."
Downing Street confirmed that David Cameron and Ireland's Taoiseach Enda Kenny had spoken earlier in the day to discuss the situation surrounding Mr Adams' arrest, but would not give any details of the call.
Mr Adams left Antrim police station through a rear exit as angry loyalist protesters waved Union flags and staged a sit-down protest in front of the fortified station.
A file will be sent to prosecutors by detectives who have questioned him over the murder of Mrs McConville.
It means the ultimate decision on whether to charge the 65-year-old politician with any offence will be made at a later date by Northern Ireland's Public Prosecution Service (PPS) after reviewing evidence presented by police.
Mr Adams has always vehemently denied allegations levelled by former republican colleagues that he ordered the murder of the 37-year-old widow in 1972.
Mr Adams, who read his statement in Irish before reading it in English, thanked everyone for the support he had been shown, adding: "I am conscious that there is another family at the heart of all of this and that is the family of Jean McConville.
"Let me be very clear. I am innocent of any involvement in any conspiracy to abduct, kill or bury Mrs McConville.
"I have worked hard with others to have this injustice redressed and for the return of the bodies of others killed during the conflict and secretly buried by the IRA, and I will continue to do so."
The Sinn Fein president said he voluntarily went to Antrim Barracks on Wednesday, having told the PSNI two months ago he was available to meet them.
He said his willingness to speak to police was "following yet another spate of media speculation - part of a sustained, malicious, untruthful and sinister campaign alleging involvement by me in the killing of Mrs Jean McConville".
He added: "When the PSNI contacted my solicitor on Monday afternoon I was concerned about the timing, given that Sinn Fein is involved in a very important European election and local government elections."
Meanwhile Mrs McConville's son Michael has called for an independent investigation into her death by a team from outside Northern Ireland to ensure there is "no political pressure".
He described how as an 11-year-old boy he saw his mother be dragged from her home in Divis flats in west Belfast by neighbours whom he recognised. But, he said, he was afraid to give their names in case he or his family are shot.
He said: "We would like to see all the investigations taken out of Northern Ireland. We would like an independent body to do this so there is no political pressure on the police."
Mr McConville also vowed his family's fight for justice would go on and said recent days had been difficult and stressful.
"The McConville family is going to stay to the bitter end of this till we get justice for our mother.
"We know it is going to be a long road but we have already been fighting for justice for 40 years and we are not going to stop now."
He expressed confidence in the police. "We know in the last while back that there's been political people pushing for people to be let go and we think there has been interference in different cases and we don't want any interference in this case by any political party."
He added: "There is not a day goes past that we don't think about our mother and this week has been particularly painful for us all.
"We all want justice for our mother and the legal process will continue as I understand that papers are being sent to the Public Prosecution Service following the questioning of Mr Adams.
"We are also glad that there has been a worldwide focus on our mother's cruel and inhuman treatment by the IRA."