Howell ruled after holding a sealed hearing on the issue.
For procedural reasons, Kamenar said, Miller needed to be held in contempt in order to initiate the appeal.
Mueller is probing Russian meddling in the 2016 United States presidential election, and Miller's subpoena is the latest to suggest the special counsel's probe is focusing at least in part on Stone, an early Trump backer with a reputation as an aggressive political operative.
But Mueller's subpoena of Miller and other Stone associates recently suggests he is now focusing on whether Stone may have had knowledge in advance about the damaging leaks of Democratic documents that Guccifer provided to Wikileaks in the months running up to the November 2016 presidential election.
Mueller has spent months investigating Russian meddling in the election and whether any Trump campaign aides had a hand in the foreign interference plot.
Stone, a longtime associate of President Trump's, met with a Russian offering dirt on Hillary Clinton in 2016.
Stone, who has denied any wrongdoing, has previously said that eight of his associates have been contacted by Mueller but that the special counsel's team has not contacted him.
Mueller has been focusing in recent months on Stone, who has had a decades-long relationship with Trump. Stone said that he was "highly confident she will testify truthfully if called upon to do so".
Miller's lawyer had sought to quash the subpoena.
Jason Sullivan, a social media expert who worked with Stone, has testified to the grand jury.
Mueller's outreach may mean he is focusing in part on whether Stone may have had advance knowledge of material detrimental to Trump's 2016 Democratic opponent Hillary Clinton that USA intelligence agencies have concluded was obtained by Russian Federation through hacking.
Earlier in 2018, Stone associates also said Stone claimed to have been briefed by WikiLeaks founder Julian Assange about stolen emails the organization had obtained before they were released.
Kamenar also said a prosecutor with powers as broad as Mueller's should be treated like a USA attorney and be subject to Senate confirmation.