The letter, written at the end of the 1916 Easter Rising, had a guide price of 1-1.5m euros.
It was the last official letter Pearse wrote, three days before his execution by firing squad on 3 May 1916.
Pearse wrote it from his prison cell to rebels still fighting in the Four Courts in Dublin.
It reads: "In order to prevent further slaughter of the civil population and in the hope of saving the lives of our followers, the members of the Provisional Government present at headquarters have decided on an unconditional surrender, and commandants or officers commanding districts will order their commands to lay down arms. PH Pearse, Dublin, 30th April 1916."
The details of the item on the James Adam's auction house website quotes historian Diarmaid Ferriter: "Behind the single sentence of the communication lie many layers.
"It was the last official letter Pearse wrote, three days before his execution by firing squad on the morning of May 3rd 1916 and is therefore a vital part of the archive of the newly declared republic in 1916.
"It marked the end of the 1916 rebellion and underlined the role of both the rebel leaders and the British forces in bringing the fighting to a conclusion.
"It is also a reminder of some of the difficulties associated with the rebels military strategy in 1916, especially communications, but also the success in securing certain positions in the midst of urban warfare."
Although the identity of the seller has not been revealed, he has been described by the auctioneers as non-Irish and living outside the country.
In a statement, Stuart Cole, a director at Adam's, said: "On this occasion, the reserve was not met.
"The owner, based overseas, has requested that Adam's apply for an export licence tomorrow (Thursday) to formally notify the [Irish] government that the document will be leaving Ireland and for the process to be expedited.
"The owner was saddened that the government refused to bid for the document but now feels relieved of his obligations to keep the document in Ireland."

The 38-year-old, who lifted the One-Day Cup at Lord's last weekend, is retiring at the end of the season.
Notts says the decision will allow the club to look to the future.
"Chris and I have had a chat. We agree that this a great fit for everyone as we prepare for a T20 campaign", said head coach Peter Moores.
Moores' son, 20-year-old Tom, will keep wicket for Notts as they begin their Twenty20 campaign against Yorkshire at Headingley.
"The thinking is that it allows us to plan ahead for the future, with Chris due to retire in September, yet it enables Chris to pass on his vast experience and knowledge to our players," Peter Moores added.
"He's excited about taking on this new role. It's a good opportunity for him to develop his coaching skills and allows Chris to rest and recharge his batteries after a very busy period.
"He'll be fresh and ready to captain the side when the County Championship matches restart in August."
A letter of surrender written by Irish rebel Patrick Pearse has been withdrawn from auction in Dublin after bidding stalled at 770,000 euros (Â£650,000).
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Notts Outlaws wicketkeeper Chris Read will be part of the club's coaching staff rather than play in this season's T20 Blast, which begins on Friday.