It's a bit muddy and will probably ultimately be decided by the lawyers, but...
According to the John Fahey, the president of WADA:
"I am confident and WADA is confident that the USADA acted within the WADA code, and that a court in Texas also decided not to interfere," Fahey said in a telephone interview. "They now have the right to apply a penalty that will be recognized by all WADA code countries around the world."
And according to the WADA Code, it is the responsibility of international organizations:
20.3.9 To vigorously pursue all potential anti-doping rule violations within its jurisdiction including investigation into whether Athlete Support Personnel or other Persons may have been involved in each case of doping.
20.3.12 To cooperate with relevant national organizations and agencies and other Anti-Doping Organizations.
So according to WADA, the USADA does have the right to impose penalties and the UCI is bound by the WADA Code to comply with their findings. They can, of course, appeal to The Court of Arbitration for Sport. And they almost certainly will.