The delay was requested in order to allow more time for reform of the security forces and state media.
Zimbabwe adopted a new Constitution this year in a trouble-free referendum backed by both long-serving President Robert Mugabe and his main rival, Prime Minister Morgan Tsvangirai.
However, the haste with which an election date was set has increased fears of a repeat of the violence and bloodshed that marred a 2008 vote and forced Mugabe and Tsvangirai to form a unity government.
Chief Justice Godfrey Chidyausiku did not immediately give reasons for the court's unanimous rejection of the requests, lodged after the regional South African Development Community (SADC) said a delay was needed to lay the ground for a credible vote.
"For the avoidance of doubt, elections should proceed on the 31st of July in terms of the proclamation by the President of Zimbabwe in compliance with the order of this court," Chidyausiku said.
Tsvangirai would abide by the decision even though his Movement for Democratic Change (MDC) party had been pushing hard for a delay of at least two weeks, his spokesperson said.
"Since the court has ruled, it means we have to go by that date," Tsvangirai spokesperson Luke Tamborinyoka said.
Tsvangirai, who claims that Mugabe's Zanu-PF party has cheated him in two other polls since 2000, has been pressing for access to state media for all political parties and cast-iron guarantees that security forces will not meddle in politics.
Mugabe (89) launches his election campaign on Friday in a bid to extend his 33 years in power despite worries over his health and age.
Tsvangirai (61) is set to launch his on Sunday.
Mugabe, Africa's oldest leader, has denied reports he suffers from prostate cancer or other major health problems.
He says he has cataracts, and returned home from Singapore on Sunday from what his spokesperson called a routine visit to an eye specialist.
He has been in power since independence from Britain in 1980, and his opponents say he cannot win a fair election after ruining one of the continent's most promising economies with policies such as the seizure of white-owned farms for blacks. – Reuters