Michael Adebalajo had been seeking leave to appeal against his whole-life sentence for killing Fusilier Lee Rigby.
Adebalajo and Michael Adebowale killed Rigby outside Woolwich barracks in south-east London in May 2013.
Adebowale is also arguing for a reduction in his sentence. He was jailed for a minimum of 45 years.
Last month, a report found Rigby's death could not have been prevented despite his killers appearing in seven intelligence investigations.
The report, by the Intelligence and Security Committee (ISC), revealed Adebowale had expressed his intent to murder a soldier in a "graphic and emotive" manner during an exchange with an overseas extremist in December 2012.
Adebolajo and Adebowale drove into the 25-year-old before hacking him to death.
At the time of his sentencing, Justice Sweeney said Adebolajo's was one of those "rare cases" that warranted a whole-life term.