Mr Watson has faced calls to apologise, from Lord Brittan’s brother Sir Samuel Brittan, over “unfounded accusations” after police dropped a rape inquiry.
Mr Watson said he was sorry for distress caused to the Brittan family. But, in a blog the MP said he had wanted the claims “properly investigated”.
Lord Brittan, whose career included two years as home secretary in Margaret Thatcher’s government, died in January aged 75. He had not been told there was no case for him to answer over an alleged rape in 1967.
Sir Samuel said Mr Watson “should apologise to my sister-in-law [Lady Brittan] for making unfounded accusations against my brother”.
Responding, Mr Watson admitted he should not have repeated a claim that Lord Brittan was “close to evil”.
“The choice facing anyone who is presented with testimony of this kind is whether to pass it on to the authorities and urge them to investigate or ignore it.”
He added: “I have said in the past that I am sorry for the distress Leon Brittan’s family experienced as they grieved for him. I still am.”
But he said he had been told of “multiple allegations”. He admitted he “did not and could not know if they were true” but said he thought they should be “fully investigated”.
“As the tributes flowed in from his lifelong friends, I felt for those people who claimed he abused them,” he said.
“The choice facing anyone presented with testimony of this kind is whether to pass it on to the authorities and urge them to investigate or to ignore it.
“I chose the first option. I felt it was my duty to do so.”
The Crown Prosecution Service found in July 2013 that there was not enough evidence for a prosecution over the claim Lord Brittan had raped a 19-year-old female student in 1967.
Mr Watson later called for a full review of all abuse allegations made against the peer.
Officers interviewed Lord Brittan, who had terminal cancer, but no charges were brought. Police have since said they would not have taken further action over the rape claim.
London mayor Boris Johnson discussed the case with Sir Bernard Hogan-Howe, Met commissioner. Mr Johnson said the delay in contacting Lord Brittan was “completely unacceptable”.
Tory MP Nigel Evans, himself cleared of sexual abuse, earlier said that Mr Watson had “set himself up as judge and jury”.
“Even when Leon had died, Tom Watson decided to repeat the allegations,” Mr Evans said. “It is totally unfounded.”
Former chancellor Norman Lamont said police investigations into historical abuse risked becoming a “witch-hunt”.