The Blairs say they fully support the MMR vaccination
Liberal Democrat leader Charles Kennedy made the demand after Mr Blair gave his clearest hint yet that 19-month-old Leo had received the triple vaccination.
In a statement Tony Blair also attacked newspapers for "horrible and unjustified" reporting of the ongoing controversy.
Mr Blair has frequently made vigorous defence of his family's right to privacy, but critics say he has already brought members of his family into his political career on several occasions.
"The suggestion that the government is advising parents to have the MMR jab whilst we are deliberately refraining from giving our child the treatment because we know it is dangerous, is offensive beyond belief," Mr Blair said.
Mr Kennedy suggested the comments were an admission that Mr Blair's son had been given the jab.
"They are more or less acknowledging, without being explicit, that they probably have had this particular jab administered for the baby and probably now the easiest thing would just be to put the issue to rest by just confirming that to be the case and be done with it," he said.
"It is not true that we believe the MMR vaccine to be dangerous or believe that it is better to have separate injections, or believe that it is linked to autism.
"On the contrary, the vaccine which is used throughout the world, helps prevent the spread of diseases that can, if contracted, cause very serious damage to children."
Leadership
Mr Blair said the reason he and his wife had refused to say whether their son had received the jab was to protect his privacy.
They believe that if they comment on one issue concerning one of their children they will be pressurised into commenting on everything from teenage alcohol abuse to underage sex.
But critics say Mr Blair has already actively brought members of his family into politics.
Last week his wife was involved in a news conference about the plight of women in Afghanistan and in 1999 Mr Blair said the controversial Millennium Dome would have to pass the "Euan test".
But Liberal Democrat health spokesman Dr Evan Harris said there were valid reasons for not disclosing either way whether a child had been given the jab.
He said some children could not have the vaccination because there was a history of epilepsy, viral infection, febrile fits or a high temperature at the time.
No comment
"Anyone who says they haven't had it would be giving away that information and I don't think that's fair on the child," said the MP for Oxford, West and Abingdon.
Other cabinet members, including Alan Milburn, have refused to comment on whether their children have received the MMR vaccination.
Health minister Yvette Cooper, however, has said her child has received the treatment.
Dr Ian Gibson, MP, who earlier this week said he believed ministers should show leadership on the issue.
He said: "There is a very difficult battle and argument going on in terms of protection of information and where you draw the line.
"But it is important to let the public know that there are people we trust, including the prime minister, who have made this decision.
"And that helps people, who are confused about the issue, take a decision which is very important for their children."