"The pardon undoubtedly spared Defendant from any punishment that might otherwise have been imposed," judge Susan Bolton wrote in a court document posted online by political journal Politico.
"It did not, however, 'revise the historical facts' of this case," Bolton wrote.
Lawyers for Arpaio have filed an appeal, Politico reported.
In August, Trump pardoned Arpaio, who became infamous for harsh methods of pursuing illegal immigrants and his tough-on-crime stance.
The 85-year-old was sheriff of Arizona's Maricopa County for 24 years before losing a re-election bid last year. In July he was found guilty on criminal contempt charges after ignoring a court order to stop imprisoning suspected illegal immigrants.
"It did not, however, 'revise the historical facts' of this case," Bolton wrote.
Lawyers for Arpaio have filed an appeal, Politico reported.
In August, Trump pardoned Arpaio, who became infamous for harsh methods of pursuing illegal immigrants and his tough-on-crime stance.
The 85-year-old was sheriff of Arizona's Maricopa County for 24 years before losing a re-election bid last year. In July he was found guilty on criminal contempt charges after ignoring a court order to stop imprisoning suspected illegal immigrants.