Security guards blocked the path of columnist and activist George Monbiot, who tried to make the arrest as Mr Bolton left the stage. The former ambassador - a key advisor to President George W Bush who argued strongly in favour of invading Iraq - had been giving a talk on international relations to more than 600 people at the literary festival.
Mr Monbiot was blocked by two heavily-built security guards at the end of the one-and-a-half hour appearance, before he could serve a "charge sheet" on him.
After being released by the guards the columnist - a fierce critic of the 2003 American-led invasion - made a dash through the rain-soaked tented village in a failed attempt to catch up with Mr Bolton.
A crowd of about 20 protestors, one dressed in a latex George Bush mask, chanted "war criminal" as Mr Bolton was ushered away.
Mr Monbiot said moments later he was "disappointed" that he had been blocked from making the citizen's arrest.
"This was a serious attempt to bring one of the perpetrators of the Iraq war to justice, for what is described under the Nuremberg Principles as an international crime," he said.
During Mr Bolton's talk, to a packed-out audience, Mr Monbiot had asked Mr Bolton what difference there was between him and a Nazi war criminal.
Mr Bolton said the war was legal, partly because Iraq had failed to comply with a key and binding UN resolution after the end of the Gulf War in 1991.
On the war's legality, he added: "This is not my personal opinion, this is the opinion of the entire legal apparatus of the US government."
A citizen's arrest can be carried out under certain circumstances by a member of the public, if they believe a person had carried out a crime, under the Serious and Organised Crime and Police Act 2005.
Earlier, festival director Peter Florence had said they had sought legal advice and been told carrying out such an arrest would be "completely unlawful" given the circumstances.
He said: ""The Hay Festival encourages visitors to voice their opinions, but also requires that, in their expression, they respect both the law and the speaker."
A spokesman for The Guardian, for which Mr Monbiot writes a regular column, said he was acting in a "personal capacity".