The treaty incorporates the Charter of Fundamental Rights, which covers a wide range of EU citizens' rights and will become legally binding once Lisbon enters into force.
Mr Miliband told the BBC's Andrew Marr show that the positions could be filled "relatively swiftly" once the Czech Republic had approved the Treaty.
But he ruled himself out from being considered for either job.
"I am not a candidate for that. I am not available," he said, adding that he was "committed" to his current job.
Endorsing Mr Blair for the role of EU president, he said whoever took the role needed to be on par with other world leaders and be able to "stop the traffic in Beijing and Moscow".
"I think it would be good for Britain and for Europe if Tony Blair was a candidate and was chosen," he said.
Mr Blair has been backed for the role by Gordon Brown and other European leaders although some, such as German leader Angela Merkel, are believed to be more sceptical.
Creates new post of EU president (President of European Council)
New post of High Representative for Foreign Affairs
More decisions by majority vote, rather than unanimity
Ratified by all member states except Czech Republic
Only Ireland held referendum on it - twice ('Yes' vote second time)
Took a decade of negotiations
Was intended to take effect in January 2009
Mr Blair, currently a Middle East envoy, has yet to say publicly whether he is interested in the role.
Should the former prime minister accept the role, it would effectively rule out Mr Miliband or another UK candidate from the foreign affairs post.
The Conservatives are opposed to Mr Blair becoming EU president, saying he is a divisive figure across Europe.
Should they win the next election, the Tories have pledged to hold a referendum in the UK on the treaty if it has not been ratified by all member states by that point.
However, they continue to face questions about what they will do if the treaty is, as seems likely, ratified by then with leader David Cameron promising he would not "let matters rest".
Shadow Business Secretary Ken Clarke told the BBC's Politics Show that the Tories would "cross that bridge" at the time.
He said he continued to support the party's "settled" policy on the issue, even though he, unlike most in his party, personally favours the treaty and suggested he would not "participate" in any referendum campaign.
He also hit out at the Czech president, Vaclav Klaus, who has said he will only sign the treaty if he secures an opt-out for his country from the Charter of Fundamental Rights, which covers EU citizens' rights.
Mr Clarke said the Czech president was "unpredictable" and described him as the "Boris Johnson of central Europe" in that he was both highly intelligent but "liked making mischief".
Czech fears that the treaty would make it easier for Germans expelled from Czechoslovakia after World War II to make property claims were "nonsense", he said.
On Friday, Mr Klaus said that he was satisfied that an EU proposal on the issue would address his concerns.
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