Kelly's family made clear they wanted officials to shoulder the blame for driving him to despair.
"Events over recent weeks have made David's life intolerable and all of those involved should reflect long and hard over that fact," they said in a statement read out by police.
The New York Times said Kelly, a former United Nations weapons inspector, had emailed a reporter before he went missing, describing "many dark actors playing games".
The paper's reporter believed Kelly meant officials with whom he had often disagreed over intelligence assessments.
Blair has ordered an inquiry into the death, but opposition Conservative Party leader Iain Duncan Smith said the probe should cover all aspects of how the Government argued for war.
Glenda Jackson, a former minister from Blair's Labour Party, said the Prime Minister "should bite the bullet and resign".
Blair himself stayed silent when a reporter in Japan asked: "Have you got blood on your hands, Mr Prime Minister? Are you going to resign over this?"
Police said Kelly's body was found late on Friday night at a wooded spot near one of his favourite hiking trails, with a knife and a bottle of painkillers. There was no evidence anybody else was involved, they said.
Kelly became embroiled in the row over the justification for war after the Government said he might be the source for a BBC report that said officials had "sexed up" intelligence about Iraq's suspected weapons.
He was grilled by Parliament over the affair last week and admitted speaking to journalists, but denied making allegations against Blair's officials.
His boss, Defence Minister Geoff Hoon, said the ministry had observed "proper procedures" in handling the affair.
"I am certainly aware that a great deal of effort was made by the Ministry of Defence to support Dr Kelly.
"Sadly that has not been sufficient in these circumstances," he said.
But when asked by a reporter if he felt personal responsibility, Hoon was clearly rattled.
"I, and I know many others, have spent the last 24 hours trying to replay these terrible events in order to try and see whether something could have been done to avoid this tragedy. All I can say is that I will co-operate fully with the inquiry."
The BBC report said a senior intelligence official had accused Blair's communications chief, Alastair Campbell, of ordering intelligence to be sensationalised.
The BBC last night finally confirmed that Kelly was the information source.
Some newspapers said the Government had forced Kelly to come forward to clear "spin doctor" Campbell's name.
- REUTERS
Death of a civil servant, a casualty of war
British Parliament Foreign Affairs Committee transcript:
Evidence of Dr David Kelly
Key players in the 'sexed-up dossier' affair
Herald Feature: Iraq
Iraq links and resources