Police accuse Shrien Dewani of hiring three South African hitmen to kill his Swedish wife. Dewani denies the charges.
Defense lawyer Julian Knowles said Dewani was suffering from post traumatic stress disorder.
However, Judge Howard Riddle rejected the claim of illness as a reason for Dewani's absence despite receiving a letter from a psychiatrist telling him it would be "detrimental" to the accused's health if he were to appear in court.
The judge said the 31-year-old Dewani would be required to attend his next court hearing on March 23. His full extradition hearing is due to begin May 3.
South African authorities are seeking to have Dewani taken back to the country to stand trial for the murder of Anni Dewani on Nov. 13.
The couple, both of Indian descent, had married in India two weeks before arriving in South Africa for their honeymoon.
He claims their vehicle was attacked by gunmen during a late-night tour of an impoverished neighborhood. Their cab driver, Zola Tongo, said Shrien Dewani offered 15,000 rand (about $2,100) to each person involved, but paid only 1,000 (about $145).
Tongo was convicted of kidnapping, murder, aggravated robbery and obstructing justice, and was sentenced to 18 years in prison.
The case has drawn wide attention in Britain and South Africa, where violent crimes are common but attacks on foreign tourists are rare.