The 33-year-old is accused of plotting to kill his Swedish-born Indian wife, Anni, in South Africa in November 2010 and is wanted there to stand trial.
The five-day hearing concluded at Westminster Magistrates' Court and Chief Magistrate Howard Riddle is due to make a ruling on July 24.
During the closing arguments in court today, Dewani's defence argued he was at a "delicate stage" in his recovery and called for a six-month adjournment to the extradition proceedings.
The Bristol-based businessman had been excused from appearing in court, having been diagnosed with post-traumatic stress disorder (PTSD) and depression. He is receiving treatment at a hospital near his home city.
"If he cannot get fit here after two-and-a-half years, one has to recognise given the severity of his condition that he may never get fit," said his defence lawyer Claire Montgomerie QC.
Dewani was ordered to return to South Africa in 2011, but this was successfully appealed against. Judges then ordered the chief magistrate to look into the case again.
Newly-wed Anni Dewani was shot when a taxi in which the couple were travelling in was hijacked in the Gugulethu township near Cape Town in November 2010.
Shrien and driver Zola Tongo were ejected from the car before she was driven away and killed.
Prosecutors claimed that he was a hitman hired by Dewani to kill his new wife, something that Shrien has consistently denied.
Tongo was jailed for 18 years after he admitted his part in the crime. Another accomplice, Mziwamadoda Qwabe, also pleaded guilty to charges over the murder and was handed a 25-year prison sentence.
Anni Dewani's family said they were hoping for "closure" as the extradition hearing ended here.
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