His refusal came after Tory MP Sir Nicholas Soames said he had "vilely traduced" the former home secretary.
But Mr Watson told the Commons he was "right to demand that guidelines were adhered to", and said abuse survivors had been "belittled for too long".
Lord Brittan died unaware that police had dropped a rape inquiry against him.
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, who died in January.
Defending his conduct, Mr Watson told the Commons he was right to raise the issue.
"I understand MPs feel aggrieved Leon Brittan was interviewed by the police and they are angry with my use of language but I am sure they would also agree that when someone is accused of multiple sexual crimes by numerous completely unrelated sources the police have a duty to investigate, no matter who it is," he said.
Mr Watson made reference to remarks made by Prime Minister David Cameron earlier, who urged him to "examine his conscience" over his actions, after Lord Brittan's brother called for an apology over "unfounded accusations".
"I think we all need to examine our consciences in this House, we presided over a state of affairs where children have been abused and then ignored, dismissed and then disdained.
"If anyone deserves an apology it is them," Mr Watson added, to cries of "shame" from the Conservative Party benches.
Sir Nicholas, who had demanded an apology to Lord Brittan's family in a point of order, said afterwards that Mr Watson's statement was a "non-apology" and that his conduct was "completely unacceptable".
"He has become the witch-finder general," he told BBC Radio 5 live. "I really do think we have got to bring some order to this debate which is running out of hand."
Meanwhile, BBC News has obtained a letter written by Mr Watson to the Director for Public Prosecutions Alison Saunders in April 2014 about the rape allegation against Lord Brittan.
In the letter he said he was "driven to the unpalatable conclusion that the identity of the alleged perpetrator - Leon Brittan - may in some way have influenced treatment of the case".
Mr Watson also made it clear he was writing the letter following complaints from the alleged victim over the way the case had been handled and urged Ms Saunders to look at it again.
The BBC also understands that four Conservative MPs have asked Home Affairs Select Committee chairman Keith Vaz to put Mr Watson's actions on the agenda when it meets on Tuesday.
The prime minister told LBC radio: "The House of Commons select committees are quite rightly going to ask him some questions so I'm sure he should answer those questions and examine his conscience about whether he's said enough so far."
On Friday Mr Watson admitted he should not have repeated a claim that Lord Brittan was "close to evil" and said he was sorry for causing distress to the Brittan family.
But the MP said he had wanted the claims against Lord Brittan, whose career included two years as home secretary in Margaret Thatcher's government, "properly investigated".
Following the reopening of the case last year, officers interviewed Lord Brittan, who had terminal cancer at the time, but no charges were brought.

Cardiff and Vale health board said admissions to the University Hospital of Wales will go to a different ward or a different hospital.
The board has not confirmed what the infections are.
Babies already being treated on the ward will remain there, so it has not closed completely.
Ruth Walker, the health board's executive nurse director, said: "In order to protect very vulnerable babies we will not be taking any planned or routine admissions over the coming days."
Labour deputy leader Tom Watson has refused to apologise for demanding that police investigate sex abuse claims against the late Lord Brittan.
[NEXT_CONCEPT]
The neonatal intensive care unit at a Cardiff hospital has stopped taking admissions due to a "cluster of infections".