In the aftermath of the annulled poll, opposition officials repeatedly described the election results as a fraud and claimed that Odinga, who leads the National Super Alliance (NASA), was the "legitimate" victor.
The justices said the Independent Electoral and Boundaries Commission had not verified the results before announcing them.
"(The board's) disobedience of this 28 Aug order in critical areas leaves us with no option but to accept the petitioner's claim that the (election board) system was infiltrated and compromised and. data interfered with", said Judge Philomena Mwilu.
A non-governmental organisation on Monday filed two petitions claiming judges had improper contact with lawyers for opposition leader Raila Odinga during his court challenge of Mr Kenyatta's win.
The constitution dictates the poll take place within 60 days of the Supreme Court's ruling, which is October 31.
"Since the Supreme Court delivered judgment. these threats have become more aggressive", Maraga said in a briefing with reporters outside the court, adding that individual judges and their staff have become subject to numerous threats on social media.
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Mwilu said the IEBC refused a court order demanding it open its computer servers.
The court held that Cebukati was responsible for failure to explain why poll results were not transmitted in the prescribed manner. "The [board] can not therefore be said to have verified the results", she noted.
"In a statement issued after a meeting with other members of the Judicial Service Commission, Mr Maraga said: "If leaders are exhausted of having a strong and independent Judiciary, they should call a referendum and abolish it all together".
The court annulled President Uhuru Kenyatta's re-election earlier this month saying there were irregularities and illegalities.
"As we have noted, out of 291 forms, 56 from that random sample, no, no, 56 did not have the watermark features, while one did not bear the serial numbers, and a further five were not signed at all", said Maraga.
Chief Justice David Maraga on Tuesday accused police of failing to provide sufficient protection - a charge police denied.
Outside the courtroom, Kenya's police tear gassed opposition and ruling party supporters who had gathered outside the Supreme Court and started jeering and pushing each other, threatening violence, said a witness.
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