Mr Bashir flew out of South Africa despite an order barring him from leaving while a Pretoria court decided whether to arrest him on charges issued by the International Criminal Court.
Mr Bashir was visiting Johannesburg for an African Union (AU) summit.
An ICC official said the failure to arrest Mr Bashir was “disappointing”.
The Pretoria High Court issued an order for Mr Bashir’s arrest hours after his aircraft left the country.
A South African judge, Dunstan Mlambo, meanwhile said the failure to arrest Mr Bashir had violated the country’s constitution.
However, Sudan described the attempt to arrest Mr Bashir as “lame and meaningless”.
Foreign Minister Ibrahim Ghandour told reporters at Khartoum airport that the court order was an attack on Sudanese sovereignty.
Mr Bashir arrived at the airport on Monday evening, dressed in white robes and waving a cane. He was greeted by cheering supporters.
The Sudanese leader is accused of committing war crimes, crimes against humanity and genocide during the Darfur conflict.
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Wanted Sudan leader
The South African press has been considering the repercussions of the attempt to arrest Sudanese President Omar al-Bashir, who had been attending an AU summit.
IOL News said Mr Bashir’s departure would leave “a major constitutional and diplomatic crisis and a big question mark over South Africa’s continued membership of the ICC”.
The Cape Times said the decision to invite the Sudanese president, despite his indictment by the ICC, had “exposed the fact that the AU considers the ICC largely irrelevant”.
“This has the potential to sound the death knell of the ICC,” the newspaper said, adding that the arrest “would set a precedent for other leaders on the continent who could be subject to the criminal jurisdiction of the ICC for their actions”.