Hisham Geneina had been arrested on Tuesday following his remarks in an interview published by Huffpost Arabi.
He had been a top campaign aide to former military chief-of-staff Sami Anan, whom the military also detained after he announced he would stand in next month's presidential election against incumbent Abdel Fattah al-Sisi.
Anan is accused of having declared his candidacy illegally while still a registered officer in the military, as well as incitement against the armed forces.
His detention may be renewed after 15 days as the military prosecution investigates.
In 2016, Geneina was sacked by Sisi as head of the Central Auditing Authority after being accused of exaggerating the cost of corruption in Egypt.
Geneina said the documents Anan allegedly possessed "revolve around political events and crises Egyptian society has passed through" since the January 2011 uprising which toppled veteran strongman Hosni Mubarak.
The uprising left a military group led by Anan and commander-in-chief Hussein Tantawi in control of the country.
His only remaining rival is a leader of a small party who had been a leading campaigner for Sisi until he abruptly announced his own candidacy.
Critics said he did so to save Sisi the embarrassment of being the only candidate, in a throwback to referendums held by Egyptian autocrats instead of elections.
The government has rejected such criticism and the candidate, Moussa Mostafa Moussa, insists he is a serious contender.
Sisi, a former defence minister, came to power after ousting his Islamist predecessor Mohamed Morsi following mass protests against the Muslim Brotherhood member.
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