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."
A letter of surrender written by Irish rebel Patrick Pearse has been withdrawn from auction in Dublin after bidding stalled at 770,000 euros (Â£650,000).