President Uhuru Kenyatta said the presidential re-run, which was expected to be a run-off between Odinga and himself, would go ahead as scheduled despite his rival's withdrawal, Reuters news agency reported.
The decision comes as a big surprise after Mr. Odinga and his NASA coalition successfully petitioned Kenya's Supreme Court to annul the results of the initial poll in August, in which incumbent president Uhuru Kenyatta was declared victor.
"In a constitutional democracy, we should not be debating about a free and fair election, or compliance with court orders, or accountability for breach of public trust", Odinga said during a press conference in Nairobi.
Odinga told supporters that Kenya's election commission had "stonewalled meaningful deliberations" on reforms to ensure a credible vote.
Kenya's Supreme Court later annulled the elections after Odinga filed a petition in court.
"For over five weeks the commission has engaged us in a ping-pong game", said NASA chief campaigner Musalia Mudavadi.
Kenya introduced electronic transmission of results in the 2013 elections to curb rigging after the flawed polls of 2007 sparked off ethnic violence that left more than 1,000 people dead and 600,000 evicted from their homes.
Odinga's opposition alliance is threatening to boycott the October vote unless the electoral board changes some personnel - a stance he declined on Monday to clarify, in comments that suggested he was keeping his options open for now.
On October 10 he withdraws from the re-run, saying the election panel has failed to make needed reforms. Police are investigating the shooting and hit and run incident, said Nairobi Central chief Harrison Thuku.
A human rights watchdog has said that 37 people were killed after Kenya's August elections, and all but two of the cases were caused by "excessive" force by police..
They said they based their 12-point irreducible minimum requirements for a fresh election on the ruling by the top court.
Meanwhile, on Monday, hundreds of opposition supporters took to the streets of Nairobi calling for electoral officials to be sacked.
Prior to the annulment, incumbent president Uhuru Kenyatta had been declared victor of the election.
The Nairobi-based court said the electoral board committed "irregularities and illegalities" during the vote, harming the integrity of the election.