Athletes affected include tennis players Venus and Serena Williams and teenage gymnast Simone Biles.
A group calling itself "Fancy Bears" claimed responsibility for the hack of a Wada database.
After the leak, Ms Biles said she had long been taking medicine for Attention Deficit Hyperactivity Disorder.
The hacker group had accused her of taking an "illicit psycho-stimulant", but she said she had "always followed the rules".
The Rio Olympics quadruple gold medallist had obtained the necessary permission to take prescription medicine on the Wada banned drugs list, USA Gymnastics said in a statement.
Wada said in a statement that the cyber attacks were an attempt to undermine the global anti-doping system.
Russian government spokesman Dmitry Peskov said it was "out of the question" that the Kremlin or secret services were involved in the hacking, Russian news agencies reported.
The hackers accessed records detailing "Therapeutic Use Exemptions" (TUEs), which allow the use of banned substances due to athletes' verified medical needs.
"By virtue of the TUE, Biles has not broken any drug-testing regulations, including at the Olympic Games in Rio," USA Gymnastics said.
Fancy Bears said TUEs amount to "licences for doping".
Russia's track and field team were banned from the Rio Olympics over an alleged state-backed doping programme. All of its athletes are barred from the ongoing Paralympics.
"Let it be known that these criminal acts are greatly compromising the effort by the global anti-doping community to re-establish trust in Russia," Wada director-general Olivier Niggli said.